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 4 of 8: Use reflection symmetry

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\alpha=\beta\implies RT=ST.

Detailed analysis

When the two angles are equal, reflection in the symmetry axis of the equilateral figure exchanges the constructions and sends R to S while fixing T. Thus RT=ST; the remaining condition for TRS to be equilateral is the angle at T being 60 degrees.