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 7 of 8: Solve the equal-angle equation

Previous step Next step

\alpha=\beta,\qquad \angle TAR=\angle TBR\implies \angle TAR=\angle TBR=15^\circ.

Detailed analysis

The equality alpha=beta makes the two displayed angles equal by the same symmetry. Their sum is 30 degrees, so each is 15 degrees; these are exactly alpha and beta in the original configuration.