## Saturday, February 21, 2015

### Geometry Problem 1088: Triangles, Perpendicular, Concurrent Lines

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Click the diagram below to enlarge it. 1. From A₁A₂BB₃ concyclic, we have ∠C₁A₁O₁=∠A₂BB₃.
From A₁A₂CC₃ concyclic, we have ∠B₁A₁O₁=180°−∠A₂CC₃.
From B₁B₂AA₃ concyclic, we have ∠C₁B₁O₁=∠B₂AA₃.
From B₁B₂CC₃ concyclic, we have ∠A₁B₁O₁=∠B₂CC₃.
From C₁C₂AA₃ concyclic, we have ∠B₁C₁O₁=∠C₂AA₃.
From C₁C₂BB₃ concyclic, we have ∠A₁C₁O₁=180°−∠C₂BB₃.

Since A₁A₂, B₁B₂, C₁C₂ concurrent at O₁,
sin∠A₁B₁O₁ / sin∠C₁B₁O₁ × sin∠B₁C₁O₁ / sin∠A₁C₁O₁ × sin∠C₁A₁O₁ / sin∠B₁A₁O₁ = 1

Therefore,
sin∠B₂CC₃ / sin∠B₂AA₃ × sin∠C₂AA₃ / sin∠C₂BB₃ × sin∠A₂BB₃ / sin∠A₂CC₃ = 1

Hence, AA₃, BB₃, CC₃ concurrent at O₂.

2. http://s11.postimg.org/jzprw7z7n/pro_1088.png
Let a, a’, b, b’ and c, c’ are angles shown in the sketch
Let (UV, YW) represent angle form by rays UV and YW
We have (CB, Cz) = a …. ( A1, A2, C, C3 cocyclic)
And ( CA,Cz) =b ….. ( C, C3, B2 and B1 cocyclic)
Similarly (AC,Ax)=b’ and (Ax,AB)= c’
And (BA,By)= c and ( By,BC)= a’

Applying Ceva’s theorem in triangle A1B1C1 and rays A1A2, B1B2 and C1C2 we have .
(sin(a)/sin(a’)) x( sin(c)/sin(c’)) x ( sin(b’)/sin(b))= 1
In triangle ABC and rays Ax, By and Cz
Verifying that (sin(CB, Cz)/sin(CA, Cz)) x( sin(AC, Ax)/sin(Ax, AB)) x ( sin(BA, By)/sin(By, BC))=
(sin(a)/sin(a’)) x( sin(c)/sin(c’)) x ( sin(b’)/sin(b))= 1
So Ax, By and Cz are concurred at O2 per converse of Ceva’s theorem