MathDB

angles of lines wanted, heart by 3 semicircles constructed

Source: Iranian Geometry Olympiad 2021 IGO Elementary p3

1/25/2022

As shown in the following figure, a heart is a shape consist of three semicircles with diameters

AB

,

BC

and

AC

such that

B

is midpoint of the segment

AC

. A heart

\omega

is given. Call a pair

(P, P')

bisector if

P

and

P'

lie on

\omega

and bisect its perimeter. Let

(P, P')

and

(Q,Q')

be bisector pairs. Tangents at points

P, P', Q

, and

Q'

to

\omega

construct a convex quadrilateral

XYZT

. If the quadrilateral

XYZT

is inscribed in a circle, find the angle between lines

PP'

and

QQ'

. https://cdn.artofproblemsolving.com/attachments/3/c/8216889594bbb504372d8cddfac73b9f56e74c.png
Proposed by Mahdi Etesamifard - Iran

geometryperimeteranglessemicircle

Igo intermediate p3

Source: Intermediate p3

12/30/2021

Given a convex quadrilateral

ABCD

with

AB = BC

and

\angle ABD = \angle BCD = 90

.Let point

E

be the intersection of diagonals

AC

and

BD

. Point

F

lies on the side

AD

such that

\frac{AF}{F D}=\frac{CE}{EA}

.. Circle

\omega

with diameter

DF

and the circumcircle of triangle

ABF

intersect for the second time at point

K

. Point

L

is the second intersection of

EF

and

\omega

. Prove that the line

KL

passes through the midpoint of

CE

. Proposed by Mahdi Etesamifard and Amir Parsa Hosseini - Iran

Intermediate p3geometry

proving tangency and collinearity

Source: IGO 2021 Advanced P3

12/30/2021

Consider a triangle

ABC

with altitudes

AD, BE

, and

CF

, and orthocenter

H

. Let the perpendicular line from

H

to

EF

intersects

EF, AB

and

AC

at

P, T

and

L

, respectively. Point

K

lies on the side

BC

such that

BD=KC

. Let

\omega

be a circle that passes through

H

and

P

, that is tangent to

AH

. Prove that circumcircle of triangle

ATL

and

\omega

are tangent, and

KH

passes through the tangency point.

geometryIGOigo p3

3

Problems(3)