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.

Question

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

Sajal Agrawal · Accepted Answer

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

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.

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.