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 8 of 8: Verify existence and conclude

Previous step Next step

\boxed{\angle CBP=\angle BCQ=15^\circ}.

Detailed analysis

The official construction with both angles 15 degrees has the required acute triangles, and the preceding equal-length and angle argument makes TRS equilateral. Therefore this value exists and is the unique possibility.