Formula for finding the coordinates of the Gergonne point of a triangle

Given any triangle with sides \(a\), \(b\) and \(c\), and coordinates of vertices \((x_{1},y_{1})\), \((x_{2},y_{2})\) and \((x_{3},y_{3})\). What are the coordinates of its Gergonne point?

There is an expression in terms of s,a,b,c.
First find \(T_{a}\) by section formula.
Then compute\( \frac{AG_{e}}{G_{e}T_{A}}\) by menelaus.
Use the section formula again to find the gergonne point.
The formula comes out to be \(\frac {\displaystyle \sum_{cyc} (s-b)(s-c)A}{\displaystyle \sum_{cyc} (s-b)(s-c)}\).
Here A is the complex number for the point A (this formula can easily be converted into cartesian form)

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## Comments

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TopNewestThere is an expression in terms of s,a,b,c. First find \(T_{a}\) by section formula. Then compute\( \frac{AG_{e}}{G_{e}T_{A}}\) by menelaus. Use the section formula again to find the gergonne point. The formula comes out to be \(\frac {\displaystyle \sum_{cyc} (s-b)(s-c)A}{\displaystyle \sum_{cyc} (s-b)(s-c)}\). Here A is the complex number for the point A (this formula can easily be converted into cartesian form)

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Aah, I need help. @Ishan Singh

@Aditya Kumar

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Its easy but bit tedious. What have you tried?

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Tried infinitely many methods, but led to zero conclusions :P

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@Xuming Liang

@Nihar Mahajan

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