MathDB

Let

P

and

Q

be the midpoints of non-parallel chords

k_1

and

k_2

of a circle

\omega

, respectively. Let the tangent lines of

\omega

passing through the endpoints of

k_1

intersect at

A

and the tangent lines passing through the endpoints of

k_2

intersect at

B

. Let the symmetric point of the orthocenter of triangle

ABP

with respect to the line

AB

R

and let the feet of the perpendiculars from

R

to the lines

AP, BP, AQ, BQ

R_1, R_2, R_3, R_4

, respectively. Prove that

\frac{AR_1}{PR_1} \cdot \frac{PR_2}{BR_2} = \frac{AR_3}{QR_3} \cdot \frac{QR_4}{BR_4}

Problem Statement