# RMO 2016

Let ∆ABC be scalene, with BC as the largest side. Let

D

be the foot of the altitude from

A onto side BC. Let points

K and

L be chosen on the lines AB and AC respectively,

such that

D is the midpoint of segment KL. Prove that the

points B, K, C, L are concyclic if and only if

∠BAC = 90

Note by A Former Brilliant Member
4 years, 4 months ago

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What have you tried? What do you know?

Staff - 4 years, 4 months ago

- 4 years, 4 months ago

I forgot to mention something

- 4 years, 4 months ago

i have already proved this you have to also prove the converse. that is not easy

- 4 years, 4 months ago

In your below post to calvin Lin , You had said that it is easy to prove that if B,K,C,L are concyclic THEN angle BAC=90 the converse (i.e , if angle BAC =90 then B,C,K,L are concyclic ) is not that easy.

- 4 years, 4 months ago

sorry its the opposite

- 4 years, 4 months ago

Messed up , but couldn't do it. Have you any idea ?

- 4 years, 4 months ago

Triangle LDC ~~ triangle BDK by SAS similarity criteria .

- 4 years, 4 months ago

it is easy to prove that angle BAC=90 if B,K,C,L are concyclic the converse is not that easy.

- 4 years, 4 months ago