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 6 of 8: Use the cyclic quadrilateral

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\angle TAR+\angle TBR=30^\circ.

Detailed analysis

In the cyclic quadrilateral, opposite angles are supplementary. Resolving the angles at A and B using the 30-degree angles of the equilateral triangle gives this sum.