MathDB

Let

ABC

be a triangle with orthocenter

H

and circumcenter

O

. Let

P

be a point in the plane such that

AP \perp BC

. Let

Q

and

R

be the reflections of

P

in the lines

CA

and

AB

, respectively. Let

Y

be the orthogonal projection of

R

onto

CA

. Let

Z

be the orthogonal projection of

Q

onto

AB

. Assume that

H \neq O

and

Y \neq Z

. Prove that

YZ \perp HO

.
[asy] import olympiad; unitsize(30); pair A,B,C,H,O,P,Q,R,Y,Z,Q2,R2,P2; A = (-14.8, -6.6); B = (-10.9, 0.3); C = (-3.1, -7.1); O = circumcenter(A,B,C); H = orthocenter(A,B,C); P = 1.2 * H - 0.2 * A; Q = reflect(A, C) * P; R = reflect(A, B) * P; Y = foot(R, C, A); Z = foot(Q, A, B); P2 = foot(A, B, C); Q2 = foot(P, C, A); R2 = foot(P, A, B); draw(B--(1.6*A-0.6*B)); draw(B--C--A); draw(P--R, blue); draw(R--Y, red); draw(P--Q, blue); draw(Q--Z, red); draw(A--P2, blue); draw(O--H, darkgreen+linewidth(1.2)); draw((1.4*Z-0.4*Y)--(4.6*Y-3.6*Z), red+linewidth(1.2)); draw(rightanglemark(R,Y,A,10), red); draw(rightanglemark(Q,Z,B,10), red); draw(rightanglemark(C,Q2,P,10), blue); draw(rightanglemark(A,R2,P,10), blue); draw(rightanglemark(B,P2,H,10), blue); label("

\textcolor{blue}{H}

",H,NW); label("

\textcolor{blue}{P}

",P,N); label("

A

",A,W); label("

B

",B,N); label("

C

",C,S); label("

O

",O,S); label("

\textcolor{blue}{Q}

",Q,E); label("

\textcolor{blue}{R}

",R,W); label("

\textcolor{red}{Y}

",Y,S); label("

\textcolor{red}{Z}

",Z,NW); dot(A, filltype=FillDraw(black)); dot(B, filltype=FillDraw(black)); dot(C, filltype=FillDraw(black)); dot(H, filltype=FillDraw(blue)); dot(P, filltype=FillDraw(blue)); dot(Q, filltype=FillDraw(blue)); dot(R, filltype=FillDraw(blue)); dot(Y, filltype=FillDraw(red)); dot(Z, filltype=FillDraw(red)); dot(O, filltype=FillDraw(black)); [/asy]

Problem Statement