Consider five points on a circle. For every three of them, we draw the perpendicular from the centroid of the triangle they determine to the line through the remaining two points. Prove that the ten lines thus formed are concurrent.

Let be an acute scalene triangle. Let and be two distinct interior points of the segment such that . Suppose that:
and are the feet of the perpendiculars from from to the lines and respectively.
and are the feet of the perpendiculars from to the lines and respectively.
Prove that and intersect on the line .

Let be a triangle with incenter , and let be a point on side . Points and are chosen on lines and respectively such that is a parallelogram. Points and are chosen on side such that and are the angle bisectors of angles and respectively. Let be the circle tangent to segment , the extension of past , and the extension of past . Prove that is tangent to the circumcircle of triangle .

A line through the vertex of the triangle which doesn't coincide with or intersectes the altitudes from and at and respectively. Let be the reflection of in and be the reflection of in Prove that the circles and are tangent.

Let be positive real numbers such that Prove that we have

Also determine the case of equality.

In triangle , points are constructed as follows: Points and lie on the extension of side such that . Points and lie on the extension of side such that . Points and lie on the extension of side such that .

The points are placed in a clockwise order around triangle .

Prove that:

Point lies inside of scalene triangle with incenter such that
Lines and intersect line respectively at and . Line through parallel to intersects at and line through parallel to intersects at . Prove that triangles and are similar.

is the -excenter of the triangle and is the circumcircle of this triangle. is the middle of arc of which doesn't contain . meets at . Prove that

Inside triangle , point is marked. Point is on segment , and point is on segment such that the circumcircles of triangles and are tangent to line . Lines are drawn through points and passing through the center of the circumcircle of triangle , and through points and passing through the center of the circumcircle of triangle . Prove that there exists a circle tangent to all four drawn lines.

Dear Mathlinkers,

1. ABC a triangle
2. D, E two point on the segment BC so that BD = DE= EC
3. M, N the midpoint of ED, AE
4. H the orthocenter of the acutangle triangle ADE
5. 1, 2 the circumcircle of the triangle DHM, EHN
6. P, Q the second point of intersection of 1 and BM, 2 and CN
7. U, V the second points of intersection of 2 and MN, PQ.

Prove : UV is parallel to PM.

Sincerely
Jean-Louis