Problem 3

Let

ABC

be an equilateral triangle. Let

P

be a point on

AC

and

Q

a point on

AB

so that triangles

ABP

and

ACQ

are acute. Let

R

and

S

be the orthocentres of triangles

ABP

and

ACQ

, respectively. Let

T

be the common point of the segments

BP

and

CQ

. Find all possible values of

\angle CBP

and

\angle BCQ

such that triangle

TRS

is equilateral.

Step 5 of 8: Convert the equilateral condition

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\angle RTA=30^\circ\iff T,B,R,A\text{ are concyclic}.

Detailed analysis

The official angle chase uses the altitude through B and the fact that ABC is equilateral. It shows that the 60-degree condition is equivalent to angle RTA=30 degrees, which is equivalent to the four points T,B,R,A being concyclic.