MathDB

Splitting quadrilateral into equal triangles

Source: Sharygin First Round 2013, Problem 5

4/8/2013

Four segments drawn from a given point inside a convex quadrilateral to its vertices, split the quadrilateral into four equal triangles. Can we assert that this quadrilateral is a rhombus?

geometryrhombusgeometry proposed

altitude, median and angle bisector concurrent

Source: Sharygin 2013 Final 8.5

8/16/2018

The altitude

AA'

, the median

BB'

, and the angle bisector

CC'

of a triangle

ABC

are concurrent at point

K

. Given that

A'K = B'K

, prove that

C'K = A'K

.

geometryangle bisectorconcurrentmedianaltitude

equal ratios on 2 sides

Source: Sharygin 2013 Final 9.5

8/19/2018

Points

E

and

F

lie on the sides

AB

and

AC

of a triangle

ABC

. Lines

EF

and

BC

meet at point

S

. Let

M

and

N

be the midpoints of

BC

and

EF

, respectively. The line passing through

A

and parallel to

MN

meets

BC

at point

K

. Prove that

\frac{BK}{CK}=\frac{FS}{ES}

. .

ratiogeometrymidpoint

Prove the collinearity

Source: 10.5 Final Round of Sharygin geometry Olympiad 2013

8/8/2013

Let ABCD is a cyclic quadrilateral inscribed in

(O)

.

E, F

are the midpoints of arcs

AB

and

CD

not containing the other vertices of the quadrilateral. The line passing through

E, F

and parallel to the diagonals of

ABCD

meet at

E, F, K, L

. Prove that

KL

passes through

O

.

geometryrhombuscyclic quadrilateralperpendicular bisectorgeometry proposed

5

Problems(4)