# Difference between revisions of "Viviani's theorem"

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[http://www.artofproblemsolving.com/Forum/viewtopic.php?type=alcumus&id=41565 Alcumus Problem] | [http://www.artofproblemsolving.com/Forum/viewtopic.php?type=alcumus&id=41565 Alcumus Problem] | ||

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[[Category:Theorems]] | [[Category:Theorems]] |

## Revision as of 12:24, 30 May 2019

The **Viviani's Theorem** states that for an equilateral triangle, the sum of the altitudes from any point in the triangle is equal to the altitude from a vertex of the triangle to the other side.

## Proof

Let be an equilateral triangle and be a point inside the triangle. We label the altitudes from to each of sides , and , and respectively. Since is equilateral, we can say that . Therefore, , and . Since the area of a triangle is the product of its base and altitude, we also have . However, the area of can also be expressed as . Therefore, , so , which means the sum of the altitudes from any point within the triangle is equal to the altitude from the vertex of a triangle.

## Problem

Alcumus Problem
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