Geometry (Thailand Math POSN 2nd round)

Write a full solution.

  1. Let HH be orthocenter of ABC\triangle ABC and point I,J,KI,J,K be a midpoint between each vertex A,B,CA,B,C and HH respectively. If PP is a point on circumcircle of ABC\triangle ABC not on the vertex A,B,CA,B,C, and MM be the midpoint between P,HP,H. Prove that I,J,K,MI,J,K,M lie on the same circle.

  2. Let the tangents of circumcircle of ABC\triangle ABC at point B,CB,C intersect at point DD. Prove that AD\overline{AD} is symmedian line of ABCABC.

  3. Let P,QP,Q be 2 points that are isogonal conjugate to each other in ABC\triangle ABC. If PP1\overline{PP_{1}} and QQ1\overline{QQ_{1}} is perpendicular to BC\overline{BC} at point P1P_{1} and Q1Q_{1} respectively, PP2\overline{PP_{2}} and QQ2\overline{QQ_{2}} is perpendicular to CA\overline{CA} at point P2P_{2} and Q2Q_{2} respectively, and PP3\overline{PP_{3}} and QQ3\overline{QQ_{3}} is perpendicular to AB\overline{AB} at point P3P_{3} and Q3Q_{3} respectively. Prove that P1,P2,P3,Q1,Q2,Q3P_{1},P_{2},P_{3},Q_{1},Q_{2},Q_{3} lie on the same circle, and the center of that circle lies in the midpoint of PP and QQ.

  4. Let I,N,HI,N,H be the center of incircle, nine-point circle, and orthocenter of ABC\triangle ABC respectively. Construct ID,NM\overline{ID}, \overline{NM} perpendicular to BC\overline{BC} at point D,MD,M respectively. If AH\overline{AH} intersect circumcircle of ABC\triangle ABC at point KK such that YY is a midpoint of AK\overline{AK}. Prove that IDNM=rAY2|ID - NM| = \displaystyle \left |r - \displaystyle \frac{AY}{2}\right| where rr is an inradius of ABC\triangle ABC.

  5. Let UU be a foot of altitude of ABC\triangle ABC from point AA. If U,UU',U'' are reflection of UU by CA,AB\overline{CA},\overline{AB} respectively, such that UU\overline{U'U''} intersect CA,AB\overline{CA},\overline{AB} at point V,WV,W respectively. Prove that BV,CW\overline{BV},\overline{CW} is perpendicular to CA,AB\overline{CA},\overline{AB} respectively.

This note is a part of Thailand Math POSN 2nd round 2015.

Note by Samuraiwarm Tsunayoshi
6 years, 4 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link]( link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

The first question can be solved using homothety. The points A,B,C and P lie on the circumcircle. So consider a homothetic transformation of the circumcircle about orthocenter and shrink the circle to half of its radius. The mid points of H and A,B,C, P will lie on this circle. This is the nine point circle of the triangle.

Pranav Rao - 5 years, 7 months ago

Log in to reply

Hi Pranav I do I know you ? I know a Pranav Rao and he is from mumbai too. This is shrihari.

Shrihari B - 5 years, 7 months ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...