MathDB

In the following, the point of intersection of two lines

g

and

h

will be abbreviated as

g\cap h

. Suppose

ABC

is a triangle in which \angle A \equal{} 90^{\circ} and

\angle B > \angle C

. Let

O

be the circumcircle of the triangle

ABC

. Let

l_{A}

and

l_{B}

be the tangents to the circle

O

A

and

B

, respectively. Let BC \cap l_{A} \equal{} S and AC \cap l_{B} \equal{} D. Furthermore, let AB \cap DS \equal{} E, and let CE \cap l_{A} \equal{} T. Denote by

P

the foot of the perpendicular from

E

l_{A}

. Denote by

Q

the point of intersection of the line

CP

with the circle

O

(different from

C

). Denote by

R

be the point of intersection of the line

QT

with the circle

O

(different from

Q

). Finally, define U \equal{} BR \cap l_{A}. Prove that \frac {SU \cdot SP}{TU \cdot TP} \equal{} \frac {SA^{2}}{TA^{2}}.

20

Problems(1)