[$XYZ$] denotes the area of $\triangle XYZ$ We have a $\triangle ABC$,$BC=6,CA=7,AB=8$ (1)If $O$ is the circumcenter of $\triangle ABC$, find [$OBC$]:[$OCA$]:[$OAB$] (2)If $H$ is the orthocenter of $\triangle ABC$, find [$HBC$]:[$HCA$]:[$HAB$]
Problem
Source: 2020 Taiwan APMO Preliminary
Tags: geometry
Cindy.tw
23.07.2020 10:27
It's trivial under the view of barycentric coordinate, however, I didn't solve it at the time.
hsiangshen
23.07.2020 13:54
Cindy.tw wrote: It's trivial under the view of barycentric coordinate, however, I didn't solve it at the time. You can use ordinary method to bash it.lol.
hsiangshen
23.07.2020 18:45
lllqy wrote: 中文俗称“奔驰定理” Nice collection.
franzliszt
10.11.2020 18:12
Use barycentrics w.r.t. $\triangle ABC$ so that $A=(1,0,0),B=(0,1,0),C=(0,0,1)$. Then, if $BC=a,CA=b,AB=c$, we obtain $$H=(\tan A:\tan B:\tan C)=((a^2+b^2-c^2)(c^2+a^2-b^2):(b^2+c^2-a^2)(a^2+b^2-c^2):(c^2+a^2-b^2)(b^2+c^2-a^2))$$and $$O=(\tan 2A:\tan 2B:\tan 2C)=(a^2(b^2+c^2-a^2):b^2(c^2+a^2-b^2):c^2(a^2+b^2-c^2))$$and the rest is a matter of patience.