Synthetic Geometry Group - Karthik's Proposal

Here are my submissions :

1) Let $ \Delta ABC $ have cevians $ \overline{AD} $ and $ \overline{CE} $, which meet at a point $ F $ inside the triangle. Prove that $ [\Delta ABC] \cdot [\Delta DEF] = [\Delta BDE] \cdot [\Delta AFC] $, where $ [ A ] $ denotes area of figure $ A $.

2) If a cevian $\overline{AQ}$ of an equilateral triangle $ABC$ is extended to meet the circumcircle at $P$ , prove that $\dfrac{1}{\overline{PB}} + \dfrac{1}{\overline{PC}} = \dfrac{1}{\overline{PQ}}$ .

#Geometry #Area #Cevians #IAmNotPro #ElementaryProblems

Easy Math Editor

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Question 1: Extend segment BF to hit AC at T. Construct $DE$ and call its intersection point with $BE$ , $M$ . Notice that the desired relation is equivalent to $\frac{[DEF]}{[BDE]} = \frac{[AFC]}{[ABC]} \implies \frac{FM}{MB} = \frac{TF}{TB}$ . We will use mass points. Put weights of $p, q, r$ on $A, B, C$ respectively. This implies that $T$ has a mass of $p+r$ with implies that $\frac{TF}{BF} = \frac{q}{p+r} \implies \frac{TF}{TB} = \frac{q}{p+q+r}$

Notice that $F$ has mass $p+q+r$ . This implies that $\frac{MF}{MB} = \frac{q}{p+q+r}$ and we are done.

Alan Yan - 5 years, 1 month ago

Nice use of Barycentric Coordinates !

Karthik Venkata - 5 years, 1 month ago

Question 2: We know by Ptomely's, that $AP = PB + PC$ .

Then we just need to prove that $\frac{1}{PB} + \frac{1}{PC} = \frac{AP}{PB \cdot PC} = \frac{1}{PQ} \implies \frac{BP}{AP} = \frac{PQ}{CP}$

Observe that $\angle ABP = \angle ACB = 60^{\circ}$ and $\angle APC = \angle ABC = 60^{\circ}$ because they substend the same arc. Similarly, $\angle CBP = \angle CAP$ substend the same arc. This implies that $\triangle BPQ \sim APC$ and the desired ratio is given with these proportional lengths.

Alan Yan - 5 years, 1 month ago

@Alan Yan Do check problem no. 5 in RMO board-2 by Nihar Mahajan ! It's more interesting than these :). Link

Karthik Venkata - 5 years, 1 month ago

@Xuming Liang @Calvin Lin @Sualeh Asif Here are some simple problems ! Presently not very good at Geometry, but planning to improve.

Karthik Venkata - 5 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}`	$\boxed{123}$