Let $ABCD$ be a convex quadrilateral. The bisector of the angle $ACD$ intersects $BD$ at point $E$. It is known that $\angle CAD = \angle BCE= 90^o$. Prove that the $AC$ is the bisector of the angle $BAE$ . (Nikolay Nikolay)
Kyiv City MO Seniors 2003+ geometry
Let $x_1, x_2, x_3, x_4$ be the distances from an arbitrary point inside the tetrahedron to the planes of its faces, and let $h_1, h_2, h_3, h_4$ be the corresponding heights of the tetrahedron. Prove that $$\sqrt{h_1+h_2+h_3+h_4} \ge \sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}$$ (Dmitry Nomirovsky)
Let the points $M$ and $N$ in the triangle $ABC$ be the midpoints of the sides $BC$ and $AC$, respectively. It is known that the point of intersection of the altitudes of the triangle $ABC$ coincides with the point of intersection of the medians of the triangle $AMN$. Find the value of the angle $ABC$.
Given a triangle $ABC$, in which $\angle B> 90^o$. Perpendicular bisector of the side $AB$ intersects the side $AC$ at the point $M$, and the perpendicular bisector of the side $AC$ intersects the extension of the side $AB$ beyond the vertex $B$ at point $N$. It is known that the segments $MN$ and $BC$ are equal and intersect at right angles. Find the values of all angles of triangle $ABC$.
Given a rectangular parallelepiped $ABCDA_1B_1C_1D_1$. Let the points $E$ and $F$ be the feet of the perpendiculars drawn from point $A$ on the lines $A_1D$ and $A_1C$, respectively, and the points $P$ and $Q$ be the feet of the perpendiculars drawn from point $B_1$ on the lines $A_1C_1$ and $A_1C$, respectively. Prove that $\angle EFA = \angle PQB_1$
In a right triangle $ABC $ with a right angle $\angle C $, n the sides $AC$ and $AB$, the points $M$ and $N$ are selected, respectively, that $CM = MN$ and $\angle MNB = \angle CBM$. Let the point $K$ be the projection of the point $C $ on the segment $MB $. Prove that the line $NK$ passes through the midpoint of the segment $BC$. (Alex Klurman)
A circle touches the sides $AC$ and $AB$ of the triangle $ABC $ at the points ${{B}_ {1}} $ and ${{C}_ {1}}$ respectively. The segments $B {{B} _ {1}} $ and $C {{C} _ {1}}$ are equal. Prove that the triangle $ABC $ is isosceles. (Timoshkevich Taras)
A circle $\omega$ is inscribed in the acute-angled triangle $\vartriangle ABC$, which touches the side $BC$ at the point $K$. On the lines $AB$ and $AC$, the points $P$ and $Q$, respectively, are chosen so that $PK \perp AC$ and $QK \perp AB$. Denote by $M$ and $N$ the points of intersection of $KP$ and $KQ$ with the circle $\omega$. Prove that if $MN \parallel PQ$, then $\vartriangle ABC$ is isosceles. (S. Slobodyanyuk)
Let $O$ be the center of the circle $\omega$ circumscribed around the acute-angled triangle $\vartriangle ABC$, and $W$ be the midpoint of the arc $BC$ of the circle $\omega$, which does not contain the point $A$, and $H$ be the point of intersection of the heights of the triangle $\vartriangle ABC$. Find the angle $\angle BAC$, if $WO = WH$. (O. Clurman)
The points $ P, Q$ are given on the plane, which are the points of intersection of the angle bisector $AL$ of some triangle $ABC$ with an inscribed circle, and the point $W$ is the intersection of the angle bisector $AL$ with a circumscribed circle other than the vertex $A$. a) Find the geometric locus of the possible location of the vertex $A$ of the triangle $ABC$. b) Find the geometric locus of the possible location of the vertex $B$ of the triangle $ABC$.
The points $A$ and $P$ are marked on the plane. Consider all such points $B, C $ of this plane that $\angle ABP = \angle MAB$ and $\angle ACP = \angle MAC $, where $M$ is the midpoint of the segment $BC$. Prove that all the circumscribed circles around the triangle $ABC$ for different points $B$ and $C$ pass through some fixed point other than the point $A$. (Alexei Klurman)
Given a triangle $ABC $, $A {{A} _ {1}} $, $B {{B} _ {1}} $, $C {{C} _ {1}}$ - its chevians intersecting at one point. ${{A} _ {0}}, {{C} _ {0}} $ - the midpoint of the sides $BC $ and $AB$ respectively. Lines ${{B} _ {1}} {{C} _ {1}} $, ${{B} _ {1}} {{A} _ {1}} $and ${ {B} _ {1}} B$ intersect the line ${{A} _ {0}} {{C} _ {0}} $ at points ${{C} _ {2}} $ , ${{A} _ {2}} $ and ${{B} _ {2}} $, respectively. Prove that the point ${{B} _ {2}} $ is the midpoint of the segment ${{A} _ {2}} {{C} _ {2}} $. (Eugene Bilokopitov)
In the tetrahedron $SABC $ at the height $SH$ the following point $O$ is chosen, such that: $$\angle AOS + \alpha = \angle BOS + \beta = \angle COS + \gamma = 180^o, $$where $\alpha, \beta, \gamma$ are dihedral angles at the edges $BC, AC, AB $, respectively, at this point $H$ lies inside the base $ABC$. Let ${{A} _ {1}}, \, {{B} _ {1}}, \, {{C} _ {1}} $be the points of intersection of lines and planes: ${{A} _ {1}} = AO \cap SBC $, ${{B} _ {1}} = BO \cap SAC $, ${{C} _ {1}} = CO \cap SBA$ . Prove that if the planes $ABC $ and ${{A} _ {1}} {{B} _ {1}} {{C} _ {1}} $ are parallel, then $SA = SB = SC $. (Alexey Klurman)
In the triangle $ABC$ the angle bisectors $AL$ and $BT$ are drawn, which intersect at the point $I$, and their extensions intersect the circle circumscribed around the triangle $ABC$ at the points $E$ and $D$ respectively. The segment $DE$ intersects the sides $AC$ and $BC$ at the points $F$ and $K$, respectively. Prove that: a) quadrilateral $IKCF$ is rhombus; b) the side of this rhombus is $\sqrt {DF \cdot EK}$. (Rozhkova Maria)
A point $O$ is chosen inside the square $ABCD$. The square $A'B'C'D'$ is the image of the square $ABCD$ under the homothety with center at point $O$ and coefficient $k> 1$ (points $A', B', C', D' $ are images of points $A, B, C, D$ respectively). Prove that the sum of the areas of the quadrilaterals $A'ABB'$ and $C'CDD'$ is equal to the sum of the areas quadrilaterals $B'BCC'$ and $D'DAA'$.
The quadrilateral $ABCD$ is inscribed in a circle and has perpendicular diagonals. Points $K,L,M,Q$ are the points of intersection of the altitudes of the triangles $ABD, ACD, BCD, ABC$, respectively. Prove that the quadrilateral $KLMQ$ is equal to the quadrilateral $ABCD$. (Rozhkova Maria)
A trapezoid $ABCD$ with bases $BC = a$ and $AD = 2a$ is drawn on the plane. Using only with a ruler, construct a triangle whose area is equal to the area of the trapezoid. With the help of a ruler you can draw straight lines through two known points. (Rozhkova Maria)
On the diagonals $AC$ and $BD$ of the inscribed quadrilateral A$BCD$, the points $X$ and $Y$ are marked, respectively, so that the quadrilateral $ABXY$ is a parallelogram. Prove that the circumscribed circles of triangles $BXD$ and $CYA$ have equal radii. (Vyacheslav Yasinsky)
Inside the parallelogram $ABCD$ are the circles $\gamma_1$ and $\gamma_2$, which are externally tangent at the point $K$. The circle $\gamma_1$ touches the sides $AD$ and $AB$ of the parallelogram, and the circle $\gamma_2$ touches the sides $CD$ and $CB$. Prove that the point $K$ lies on the diagonal $AC$ of the paralelogram.
The triangle $ABC$ with $AB> AC$ is inscribed in a circle, the angle bisector of $\angle BAC$ intersects the side $BC$ of the triangle at the point $K$, and the circumscribed circle at the point $M$. The midline of $\Delta ABC$, which is parallel to the side $AB$, intersects $AM$ at the point $O$, the line $CO$ intersects the line $AB$ at the point $N$. Prove that a circle can be circumscribed around the quadrilateral $BNKM$. (Nagel Igor)
Inside the triangle $ABC$ choose the point $M$, and on the side $BC$ - the point $K$ in such a way that $MK || AB$. The circle passing through the points $M, \, \, K, \, \, C,$ crosses the side $AC$ for the second time at the point $N$, a circle passing through the points $M, \, \, N, \, \, A, $ crosses the side $AB$ for the second time at the point $Q$. Prove that $BM = KQ$. (Nagel Igor)
The two circles ${{w} _ {1}}, \, \, {{w} _ {2}}$ touch externally at the point $Q$. The common external tangent of these circles is tangent to ${{w} _ {1}}$ at the point $B$, $BA$ is the diameter of this circle. A tangent to the circle ${{w} _ {2}} $ is drawn through the point $A$, which touches this circle at the point $C$, such that the points $B$ and $C$ lie in one half-plane relative to the line $AQ$. Prove that the circle ${{w} _ {1}}$ bisects the segment $C $. (Igor Nagel)
The segment $AB$ is the diameter of the circle. The points $M$ and $C$ belong to this circle and are located in different half-planes relative to the line $AB$. From the point $M$ the perpendiculars $MN$ and $MK$ are drawn on the lines $AB$ and $AC$, respectively. Prove that the line $KN$ intersects the segment $CM$ in its midpoint. (Igor Nagel)
The altitueds $A {{A} _ {1}} $, $B {{B} _ {1}}$ and $C {C} _ 1$ are drawn in the acute triangle $ABC$. . The perpendicular $AK$ is drawn from the vertex $A$ on the line ${{A} _ {1}} {{B} _ {1}}$, and the perpendicular $BL$ is drawn from the vertex $B$ on the line ${{C} _ {1}} {{B} _ {1}}$. Prove that ${{A} _ {1}} K = {{B} _ {1}} L$. (Maria Rozhkova)
In the triangle $ABC$ the side $AC = \tfrac {1} {2} (AB + BC) $, $BL$ is the bisector $\angle ABC$, $K, \, \, M $ - the midpoints of the sides $AB$ and $BC$, respectively. Find the value $\angle KLM$ if $\angle ABC = \beta$
In the triangle $ABC$, for which $AC <AB <BC$, on the sides $AB$ and $BC$ the points $K$ and $N$ were chosen, respectively, that $KA = AC = CN$. The lines $AN$ and $CK$ intersect at the point $O$. From the point $O$ held the segment $OM \perp AC $ ($M \in AC$) . Prove that the circles inscribed in triangles $ABM$ and $CBM$ are tangent. (Igor Nagel)
Construct for the triangle $ABC$ a circle $S$ passing through the point $B$ and touching the line $CA$ at the point $A$, a circle $T$ passing through the point $C$ and touches the line $BA$ at the point $A$. The second intersection point of the circles $S$ and $T$ is denoted by $D$. The intersection point of the line $AD$ and the circumscribed circle $\Delta ABC$ is denoted by $E$. Prove that $D$ is the midpoint of the segment $AE$.
Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers at points ${{O} _ {1}}$ and ${{ O} _ {2}}$ intersect at points $A$ and $B$, respectively. Around the triangle ${{O} _ {1}} {{O} _ {2}} B$ circumscribe a circle $w$ centered at the point $O$, which intersects the circles ${{w } _ {1}}$ and ${{w} _ {2}}$ for the second time at points $K$ and $L$, respectively. The line $OA$ intersects the circles ${{w} _ {1}}$ and ${{w} _ {2}}$ at the points $M$ and $N$, respectively. The lines $MK$ and $NL$ intersect at the point $P$. Prove that the point $P$ lies on the circle $w$ and $PM = PN$. (Vadym Mitrofanov)
The points $X, \, \, Y$are selected on the sides $AB$ and $AD$ of the convex quadrilateral $ABCD$, respectively. Find the ratio $AX \, \,: \, \, BX$ if you know that $CX || DA$, $DX || CB$, $BY || CD$ and $CY || BA$.
In the acute-angled triangle $ ABC $, the sides $ AB $ and $BC$ have different lengths, and the extension of the median $ BM $ intersects the circumscribed circle at the point $ N $. On this circle we note such a point $ D $ that $ \angle BDH = 90 {} ^ \circ $, where $ H $ is the point of intersection of the altitudes of the triangle $ ABC $. The point $K$ is chosen so that $ ANCK $ is a parallelogram. Prove that the lines $ AC $, $ KH $ and $ BD $ intersect at one point. (Igor Nagel)
On the bisector of the angle $ BAC $ of the triangle $ ABC $ we choose the points $ {{B} _ {1}}, \, \, {{C} _ {1}} $ for which $ B {{B} _ {1 }}\perp AB $, $ C {{C} _ {1}} \perp AC $. The point $ M $ is the midpoint of the segment $ {{B} _ {1}} {{C} _ {1}} $. Prove that $ MB = MC $.
On the circle with diameter $AB$, the point $M$ was selected and fixed. Then the point ${{Q} _ {i}}$ is selected, for which the chord $M {{Q} _ {i}}$ intersects $AB$ at the point ${{K} _ {i}}$ and thus $ \angle M {{K} _ {i}} B <90 {} ^ \circ$. A chord that is perpendicular to $AB$ and passes through the point ${{K} _ {i}}$ intersects the line $B {{Q} _ {i}}$ at the point ${{P } _ {i}}$. Prove that the points ${{P} _ {i}}$ in all possible choices of the point ${{Q} _ {i}}$ lie on the same line. (Igor Nagel)
The median $AM$ is drawn in the acute-angled triangle $ABC$ with different sides. Its extension intersects the circumscribed circle $w$ of this triangle at the point $P$. Let $A {{H} _ {1}}$ be the altitude $\Delta ABC$, $H$ be the point of intersection of its altitudes. The rays $MH$ and $P {{H} _ {1}}$ intersect the circle $w$ at the points $K$ and $T$, respectively. Prove that the circumscribed circle of $\Delta KT {{H} _ {1}}$ touches the segment $BC$. (Hilko Danilo)
In the triangle $ABC$ the angle bisector $AD$ is drawn, $E$ is the point of tangency of the inscribed circle to the side $BC$, $I$ is the center of the inscribed circle $\Delta ABC$. The point ${{A} _ {1}}$ on the circumscribed circle $\Delta ABC$ is such that $A {{A} _ {1}} || BC$. Denote by $T$ - the second point of intersection of the line $E {{A} _ {1}}$ and the circumscribed circle $\Delta AED$. Prove that $IT = IA$.
Given the square $ABCD$. Let point $M$ be the midpoint of the side $BC$, and $H$ be the foot of the perpendicular from vertex $C$ on the segment $DM$. Prove that $AB = AH$. (Danilo Hilko)
In the acute isosceles triangle $ABC$ the altitudes $BB_1$ and $CC_1$ are drawn, which intersect at the point $H$. Let $L_1$ and $L_2$ be the feet of the angle bisectors of the triangles $B_1AC_1$ and $B_1HC_1$ drawn from vertices $A$ and $H$, respectively. The circumscribed circles of triangles $AHL_1$ and $AHL_2$ intersects the line $B_1C_1$ for the second time at points $P$ and $Q$, respectively. Prove that points $B, C, P$ and $Q$ lie on the same circle. (M. Plotnikov, D. Hilko)
The bisector $AD$ is drawn in the triangle $ABC$. Circle $k$ passes through the vertex $A$ and touches the side $BC$ at point $D$. Prove that the circle circumscribed around $ABC$ touches the circle $k$ at point $A$.
In the acute-angled triangle $ABC$, the altitudes $BP$ and $CQ$ were drawn, and the point $T$ is the intersection point of the altitudes of $\Delta PAQ$. It turned out that $\angle CTB = 90 {} ^ \circ$. Find the measure of $\angle BAC$. (Mikhail Plotnikov)
Given an isosceles $ABC$, which has $2AC = AB + BC$. Denote $I$ the center of the inscribed circle, $K$ the midpoint of the arc $ABC$ of the circumscribed circle. Let $T$ be such a point on the line $AC$ that $\angle TIB = 90 {} ^ \circ$. Prove that the line $TB$ touches the circumscribed circle $\Delta KBI$. (Anton Trygub)
In the quadrilateral $ABCD$, the diagonal $AC$ is the bisector $\angle BAD$ and $\angle ADC = \angle ACB$. The points $X, \, \, Y$ are the feet of the perpendiculars drawn from the point $A$ on the lines $BC, \, \, CD$, respectively. Prove that the orthocenter $\Delta AXY$ lies on the line $BD$.
Call a right triangle $ABC$ special if the lengths of its sides $AB, BC$ and$ CA$ are integers, and on each of these sides has some point $X$ (different from the vertices of $ \vartriangle ABC$), for which the lengths of the segments $AX, BX$ and $CX$ are integers numbers. Find at least one special triangle. (Maria Rozhkova)
In an acute-angled triangle $ABC$, in which $AB<AC$, the point $M$ is the midpoint of the side $BC, K$ is the midpoint of the broken line segment $BAC$ . Prove that $\sqrt2 KM > AB$. (George Naumenko)
Given an acute isosceles triangle $ABC, AK$ and $CN$ are its angle bisectors, $I$ is their intersection point . Let point $X$ be the other intersection point of the circles circumscribed around $\vartriangle ABC$ and $\vartriangle KBN$. Let $M$ be the midpoint of $AC$. Prove that the Euler line of $\vartriangle ABC$ is perpendicular to the line $BI$ if and only if the points $X, I$ and $M$ lie on the same line. (Kivva Bogdan)
Let $\Gamma$ be a semicircle with diameter $AB$. On this diameter is selected a point $C$, and on the semicircle are selected points $D$ and $E$ so that $E$ lies between $B$ and $D$. It turned out that $\angle ACD = \angle ECB$. The intersection point of the tangents to $\Gamma$ at points $D$ and $E$ is denoted by $F$. Prove that $\angle EFD=\angle ACD+ \angle ECB$.
Circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. A point $C$ is constructed such that $AO_2CO_1$ is a parallelogram. An arbitrary line is drawn through point $A$, which intersects the circles $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y$, respectively. Prove that $CX = CY$. (Oleksii Masalitin)
Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $P$. The circumscribed circles of triangles $APD$ and $BPC$ intersect the line $AB$ at points $E, F$ correspondingly. $Q$ and $R$ are the projections of $P$ onto the lines $FC, DE$ correspondingly. Show that $AB \parallel QR$. (Proposed by Mykhailo Shtandenko)
Let $H$ and $O$ be the orthocenter and the circumcenter of the triangle $ABC$. Line $OH$ intersects the sides $AB, AC$ at points $X, Y$ correspondingly, so that $H$ belongs to the segment $OX$. It turned out that $XH = HO = OY$. Find $\angle BAC$. (Proposed by Oleksii Masalitin)