Synthetic Geometry Group-Sualeh's Proposal (Pakistan Round 1)

These problems are my first submissions to Xuming's Geometry group.

However they are for the purpose of sharing the Geometry problems from Pakistan's First Round (for 2016)

Q1.The tangents at $A,B$ to the circumcircle$\omega$ of Triangle $ABC$ meet at $T$. The line through $T||AC$ meets $BC$ at $D$. Prove that $AD=CD$.

Q2. In a right angled triangle $ABC, \hat{C}=90^{\circ}, CD\perp AB$ at $D$ and the angle bisector of $\hat{B}$ intersects $CD,AC$ at $O,E$ respectively. Through $O$ introduce $FG \parallel AB$ such that $FG$ intersects $AC,BC$ at $F,G$ respectively. Prove that $AF=CE$ .

Geometry-Pakistan's First Round (for IMO 2016).

#Geometry #AngleBisectorTheorem #Circumcircle #CyclicQuadrilaterals #MadMath

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Comments

Due to lack of time I have uploaded pics of my solution to problem 1.

Nihar Mahajan - 5 years, 8 months ago

Good job Nihar ! The crux move was to identify that TBDA is cyclic, by wishful thinking.

Venkata Karthik Bandaru - 5 years, 8 months ago

absolutely!

Sualeh Asif - 5 years, 8 months ago

@Calvin Lin @Xuming Liang Here is my submission!

@Nihar Mahajan ,@Sharky Kesa I thought you would be interested!

Sualeh Asif - 5 years, 8 months ago

Ill add in the solutions when requested. (Not before a week though)

Sualeh Asif - 5 years, 8 months ago

1st one is just some simple angle chase using alternate segment theorem as proved in the solution by @Nihar Mahajan.

2nd one is also just trivial: Solution to II.

$\Delta DBC \sim \Delta CBA$

=> $\dfrac{DB}{CB} = \dfrac{CB}{AB}$

=> $\dfrac{DO}{OC} = \dfrac{CE}{AE}$ --- (Int. Angle Bisector Theorem)

=> $\dfrac{AF}{CF} = \dfrac{CE}{AE}$

=> $\dfrac{AF}{AC} = \dfrac{CE}{AC}$ ---- (By rule of Dividendo)

=> $AF = AC$

K.I.P.K.I.G

Vishwash Kumar ΓΞΩ - 4 years, 1 month ago

Math	Appears 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}`	$\boxed{123}$