1)AD,BE and CF, the altitudes of triangle ABC are equal. prove that triangle ABC is an equilateral triangle.

2) Prove that the sum of all sides of a quadrilateral is greater than the sum of its diagnols.

AD,BE and CF, the altitudes of triangle ABC are equal. prove that triangle ABC is an equilateral triangle

We have AD , BE and CF are the altituds of a triangle ABC

Where , AD = BE= CF

Now , let the area of triangle be a.

therefore area of triangle,

* BC * AD = * AC * BE = * AB * CF = a

We know AD = BE = CF

So, * BC * AD = * AC * AD = * AB * AD

BC = AC = AB

Hence, ABC is an equilateral triangle

2) Prove that the sum of all sides of a quadrilateral is greater than the sum of its diagonals

A B

D C

Draw any quadrilateral and label the vertices A, B, C, and D with vertex A opposite vertex C and vertex B opposite D.

The sides are AB, BC, CD and DA and the diagonals are AC and BD

You are to prove AB + BC + CD + AD > AC + BD

In figure, you formed the triangles, ACD , ABC, ABD and BDC

You know that the sum of two sides of a triangle is always greater than the two sides

In ACD: AD + CD > AC

In ABC: AB + BC > AC

in ABD: AB + AD > BD

in BDC: BC + CD > BD

Adding these 4 inequalities

2AB + 2BC + 2CD + 2AD > 2AC + 2BD

Dividing by 2

AB+BC+CD+AD = AC + BD

PROVED

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