Show that the diameter of the inscribed circle of a triangle is less than the length of any side of the triangle

In an acute triangle \triangle ABC, the longest altitude BE is equal to the median CD in length. Show that \angle ACB <60\degree

In \triangle ABC, segment CD bisects \angle C and intersects AB at point D. Show that CD^2 < CA\cdotp CB

Verify that the length from any point in a right triangle to the endpoints of hypotenuse is less than the length of the hypotenuse.

Suppose \triangle ABC is a triangle with area 24 and that there is a point P inside \triangle ABC which is distance 2 from each of the sides of \triangle ABC. What is the perimeter of \triangle ABC?

In \triangle ABC , D, E are two points on the sides of AC and AB. BD and CE intersects at point O , \angle OBC=\angle OCB=\dfrac{1}{2}\angle A. verify that BE=CD.

Derivation of Golden Ratio by Geometric Shapes

In a 30-60-90 triangle, the length of the hypotenuse is 6. What is the length of the shortest side?

In figure, if \overline{ABC} is a straight line , then x =

In the figure, point D lies on the side of AC in \triangle ABC, AC=25 , AB=35. Point E and F are movable on the side of AB (point F is on the left of point E) and \angle EDF=\angle A

In the figure, is a \triangle ABC in which, \angle ACB =90\degree , AC=3 , BC = 4 , and point D , E are two points on the side of AB ( not coincide with D , E ) , and \angle DCE = \angle B .

In the figure, \triangle ABC is a right triangle in which, \angle ACB = 90\degree, AB=13, CD\parallel AB, point E is a dynamic point in the ray of CD ( not coincide with point C ) , connect AE, intersect BC at point F, \angle BAE bisects BC at point G。

In the figure, △ABC is a right triangle, where the internal angle \angle C = 90\degree , \angle A=30° , the segment AD bisects ∠A , which one is the value of the expression \dfrac{AB}{CD}-\dfrac{AC}{CD}

In the triangle ACE , AE is parallel to BD , what is the approximate length of DE ?

Stella leaves her house and starts driving due south 30 miles, then drives due west for 60 miles, and finally due north for 10 milesto reach her office.which of the following approximate straight-line distance, in miles, from her house to office?