Inequalities in a Triangle

$AD$ is a median of the triangle $ABC$. If $AB+BC+CA>x$ then find $x$.

ABC is an isosceles triangle with AB = AC. D is any point online segment BC and E is a point on BC produced.
Prove that :  AE > AD 

$ABC$ is an isosceles triangle with $AB = AC.$ $D$ is any point online segment $BC$and $E$ is a point on $BC$ produced.
Prove that :  $AE > AC$

$ABC$ is an isosceles triangle with $AB = AC$.$D$ is any point on line segment $BC$ and $E$ is a point on $BC$ produced.
Prove that:  $AC > AD$

Prove that in a triangle, other than equilateral triangle, the angle opposite to the longest side is greater than $\frac{2}{3}$ of a right angle.

 In a $△$PQR, PS is drawn perpendicular to QR. If SR > QS, Show that PR > PQ.

 If O is any point in the exterior of $△$ ABC, show that $\frac{1}{2}$ (AB+BC+AC)< (OA+OB+OC) 

 ABC is a triangle and O is any point inside the triangle. Show that OB + OC < AB + AC

In Figure , the sides $AB$ and $AC$ of $△ABC$ are produced and the bisectors of external angles $B$ and $C$ meet at $P$. If
$AB>AC$, Prove that $PC>PB$. 

In Figure, $P$ is any point inside the triangle $ABC$.  Prove that $(AP + BP +PC) =\frac{1}{2}(AB+BC+AC)$.

Can you draw a triangle with sides  $3.2 \mathrm{cm}$, $4.5 \mathrm{cm}$ and $3.2 \mathrm{cm}$?

Can you draw a triangle with sides $2.5 \mathrm{cm}, 2.5 \mathrm{cm}$ and $5.0 \mathrm{cm}$?

Can you draw a triangle with sides $3 \mathrm{cm}, 2.5 \mathrm{cm}$ and $6.0 \mathrm{cm}$?

Can you draw a triangle with sides  $4 \mathrm{cm}, 6 \mathrm{cm}$ and $7 \mathrm{cm}$?

$ABCD$ is a quadrilateral. $P$ is any point inside the quadrilateral. Prove that $PA+PB+PC+PD>AC+BD$.

A student has proposed triangles with the following dimensions:

i) 4 cm, 8 cm, 3 cm

ii)1000 meter, 1500 meter, 500 meter

iii) 2 km, 3km, 10 km

iv)1mm, 3mm, 3mm

Find which one of his proposals is correct.

Shown below are some triangles Salim constructed,△OAB, △OAC, and △OAD

Now he wants to construct one more triangle OAE, such that OA + OE = AE. He marks point E on the circle and joined E with points A and O.

Which of the following constructions could he have made?

In a triangle $ABC$, median $AD, BE \mathrm{and} CF$ intersect at point $G$ prove that: $4\left(AD+BE+CF\right)>3\left(AB+BC+CA\right)$

Two sides of a triangle are of lengths $5 \mathrm{c}\mathrm{m}$ and $1.5 \mathrm{c}\mathrm{m}$. The length of the third side cannot be more than or equal to $x \mathrm{cm}$. Find the value of $x$.

In $△\mathrm{ABC}$, $\angle \mathrm{A}:\angle \mathrm{B}:\angle \mathrm{C}=5:6:7$. Without finding the angles of the triangles, name its smallest side in alphabetical order.