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Hip1zzzil   7
N 32 minutes ago by sansae
Source: FKMO 2025 P3
An acute triangle $\bigtriangleup ABC$ is given which $BC>CA>AB$.
$I$ is the interior and the incircle of $\bigtriangleup ABC$ meets $BC, CA, AB$ at $D,E,F$. $AD$ and $BE$ meet at $P$. Let $l_{1}$ be a tangent from D to the circumcircle of $\bigtriangleup DIP$, and define $l_{2}$ and $l_{3}$ on $E$ and $F$, respectively.
Prove $l_{1},l_{2},l_{3}$ meet at one point.
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Hip1zzzil
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sansae
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Cosine Law and Leibniz's Theorem
narutomath   3
N Jul 9, 2016 by Virgil Nicula
The hypotenuse of a right triangle has length 1.The centroid of the triangle lies on the inscribed circle. Find the perimeter of the triangle.

I know a solution with Leibniz's Theorem, but the problem is in Cosine Law section. Can someone offer me a solution without Leibniz?
3 replies
narutomath
Jul 7, 2016
Virgil Nicula
Jul 9, 2016
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Cosine Law and Leibniz's Theorem
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narutomath
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narutomath
#1 h  Jul 7, 2016, 4:15 PM • 1 Y
Y by Adventure10
The hypotenuse of a right triangle has length 1.The centroid of the triangle lies on the inscribed circle. Find the perimeter of the triangle.

I know a solution with Leibniz's Theorem, but the problem is in Cosine Law section. Can someone offer me a solution without Leibniz?
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suli
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suli
#2 h  Jul 7, 2016, 4:31 PM • 2 Y
Y by Adventure10, Mango247
I will offer my solution if you can tell me what is Leibniz's theorem.

Solution
This post has been edited 1 time. Last edited by suli, Jul 7, 2016, 4:33 PM
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george_54
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george_54
#3 h  Jul 7, 2016, 4:39 PM • 1 Y
Y by Adventure10
suli wrote:
I will offer my solution if you can tell me what is Leibniz's theorem.

If $G$ is the centroid of a triangle $ABC$ and $M$ is any point of the plane, then:

$\boxed{M{A^2} + M{B^2} + M{C^2} = 3M{G^2} + \frac{1}{3}({a^2} + {b^2} + {c^2})}$
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Virgil Nicula
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Virgil Nicula
#4 h  Jul 9, 2016, 3:24 AM • 2 Y
Y by Adventure10, Mango247
See PP12 from here and PP19 from here.
This post has been edited 1 time. Last edited by Virgil Nicula, Jul 9, 2016, 4:38 PM
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