In a triangle $ABC$, $\angle B=90^o$ and $\angle A=60^o$, $I$ is the point of intersection of its angle bisectors. A line passing through the point $I$ parallel to the line $AC$, intersects the sides $AB$ and $BC$ at the points $P$ and $T$ respectively. Prove that $3PI+IT=AC$ . (Anton Trygub)
Problem
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Tags: geometry, incenter, right triangle
natmath
15.02.2021 00:16
WLOG $BC=1$
Note that $\Delta BPT\sim BAC$, so we can send $\Delta BAC$ to $\Delta BPT$ with a homothety centered around $B$. This homothety has ratio $\frac{h_B-r}{h_B}$. Where $h_B$ is the altitude of $\Delta BAC$ from $B$ and $r$ is the inradius of $\Delta BAC$.
We can easily get
$$r=\frac{\sqrt{3}-1}{2}$$$$h_B=\frac{\sqrt{3}}{2}$$So the homothety has ratio $\frac{1}{\sqrt{3}}$
Denote the foot of the angle bisector of $B$ onto $AC$ as point $D$. Since $\angle DBC=45$ and $\angle BDC=75$, by LOS we have
$$DC=\frac{\sin 75}{\sin 45}=\frac{1+\sqrt{3}}{2}$$Note that $DC$ is sent to $IT$ and $AC$ is sent to $PT$ by the initially described homothety. So $3PI+IT=$
$$3PT-2IT$$$$\frac{1}{\sqrt{3}}(5-\sqrt{3})
@OP Are you sure this question is correct. I am getting that $AC=\sqrt{3}PT$, so obviously $AC<3PT$. I also made this geogebra sketch, which disproves the stated problem https://www.geogebra.org/calculator/hhvabw63
parmenides51
15.02.2021 00:22
incorrect original post parmenides51 wrote: In a triangle $ABC$, $\angle B=90^o$ and $\angle C=60^o$, $I$ is the point of intersection of its angle bisectors. A line passing through the point $I$ parallel to the line $AC$, intersects the sides $AB$ and $BC$ at the points $P$ and $T$ respectively. Prove that $3PT+IT=AC$ . (Anton Trygub) oops I had 2 typos (following the red ones) m the correct is parmenides51 wrote: In a triangle $ABC$, $\angle B=90^o$ and $\angle A=60^o$, $I$ is the point of intersection of its angle bisectors. A line passing through the point $I$ parallel to the line $AC$, intersects the sides $AB$ and $BC$ at the points $P$ and $T$ respectively. Prove that $3PI+IT=AC$ . (Anton Trygub) thanks for noticing