RMO 2016

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

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

#Geometry

Easy Math Editor

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Comments

What have you tried? What do you know?

Calvin Lin Staff - 4 years, 7 months ago

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

I forgot to mention something

vishwash kumar - 4 years, 7 months ago

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

A Former Brilliant Member - 4 years, 7 months ago

@A Former Brilliant Member – 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.

Vishwash Kumar ΓΞΩ - 4 years, 7 months ago

@Vishwash Kumar Γξω – sorry its the opposite

A Former Brilliant Member - 4 years, 7 months ago

@A Former Brilliant Member – Messed up , but couldn't do it. Have you any idea ?

vishwash kumar - 4 years, 7 months ago

Triangle LDC ~~ triangle BDK by SAS similarity criteria .

vishwash kumar - 4 years, 7 months ago

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

A Former Brilliant Member - 4 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$