Geometry Set

The problems are in no particular order. Please reshare the note if you like what you see, feel free to share comments or ideas and enjoy.

  1. Let \(P\) and \(Q\) be points on the side \(AB\) of the triangle \(ABC\) (with \(P\) between \(A\) and \(Q\)) such that \(\angle{ACP}=\angle{PCQ}=\angle{QCB}\), and let \(AD\) be the angle bisector of \(\angle{BAC}\) with \(D\) on \(BC\). Line \(AD\) meets lines \(CP\) and \(CQ\) at \(M\) and \(N\) respectively. Given that \(PN=CD\) and \(3\angle{BAC}=2\angle{BCA}\), prove that triangles \(CQD\) and \(QNB\) have the same area. (Belarus 1999)

  2. Let \(ABC\) be a triangle and let \(\omega\) be its incircle. Let \(D_1\) and \(E_1\) be the tangency points of \(\omega\) with sides \(BC\) and \(AC\), respectively. Let \(D_2\) and \(E_2\) be points on sides \(BC\) and \(AC\), respectively, such that \(CD_2=BD_1\) and \(CE_2=AE_1\), and let \(P\) be the intersection point of segments \(AD_2\) and \(BE_2\). Circle \(\omega\) intersects segment \(AD_2\) in two points, being the closest one to \(A\) denoted by \(Q\). Prove that \(AQ=D_2P\).

  3. Points \(A, B, C, D\) are four consecutive vertices of a regular polygon such that \[\dfrac{1}{AB}=\dfrac{1}{AC}+\dfrac{1}{AD}\]. How many sides does the polygon have?

Note by José Marín Guzmán
3 years, 6 months ago

No vote yet
1 vote

  Easy Math Editor

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 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)


Sort by:

Top Newest

3)The question number 3 is easy to solve with complex numbers.But also found a geometric solution.

Let the polygon be \(n\)- sided.

Suppose the vertex after \(D\) be \(E\).Hence the \(5\) consecutive vertices are \(A,B,C,D,E\) ( arranged anticlockwise from A to E,say)

Firstly,\(\frac{1}{AB}=\frac{1}{AC}+\frac{1}{AD}\) gives


One can easily show that quadrilateral \(ACDE\) is cyclic.

So,by Ptolemy's theorem, \(AE.CD+AC.ED=AD.EC\)...\((2)\)

Also,since the polygon is regular,we have \(AB=CD=ED\) and \(EC=AC\).

So,from \((1)\), \(AB.AD+AB.AC=AC.AD=CE.AD=AE.CD+AC.ED=AE.AB+AC.AB\)


Since,these two diagonals are equal,number of vertices between \(A\) and \(E\)(moving clockwise) equals the number of vertices between \(A\) and \(D\)(moving anti-clockwise).

Hence,\(n-5=2\) or \(n=7\).

Souryajit Roy - 3 years, 6 months ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...