Plane Geometry

For those of you not familiar with Hansen's Theorem, it is very useful when deriving the radii of the excircles of a right triangle. It states that the Sum of the squares of the radii of the three excircles of a right triangle, plus the square of the radius of its incircle is equal to the sum of the squares of the sides of the triangle.

The theorem also shows an elegant and quick method for calculating the radii of the 3 excircles of the triangle, and proves that the Sum of the 4 radii is equal to the Perimeter of the triangle.

PROPOSITION: The hypotenuses of two dissimilar right triangles “A” and “B”, are twice the legs of a known right triangle “C”, and the altitude to hypotenuses in each of A and B triangles are identical to that in C.

HYPOTHESIS: The sum of the greater segments on each of the hypotenuses of A and B, caused by the altitude to hypotenuse in C, will equal the Perimeter of C.

For instance, triangle C can be any right triangle whose 3 sides are known.

Let “m” and “n” be the long and short legs, respectively, of C.

Let "d" be the altitude to hypotenuse of C.

I need help to this problem I wonder if the answers are in the following choices. The problem is.. A vertical parabolic curve, 135 m long has a tangent grades of -1.7% and 2.3% . Find the sight distance in m.
a. 139.6 c. 147.1
b. 149.8 d. 140.6

What is the perimeter of the value of the curve i=3(1 +cos x)?

Example 016: Radius of sphere circumscribing a regular triangular pyramid

I would like to add a modest contribution to a solution 016.
Radius of inscribbed sphere can be obtained directly (without slope angle determination):

√((AE)^2+(OE)^2 ) = ED-OE → OE = r = ((ED)^2-(AE)^2)/(2*(ED) ) and so on

Best regards !

Matija Oblak

a fly stationed at a point on the circumference of the base of a cylindrical tower of radius 12 feet. Find that he can just see a distant flagstaff by walking along the tangent line to the base either a distance 8 feet in one direction or a distance 5 feet in the other direction. Find the distance of the foot of the flagstaff from the center of the base of the tower

A concrete dam of hieght 128 ft. was built in a gorge. One side AB of the gorge slopes at an angle of 60 degrees, the other side CD at 45 degrees. The bases of the dam are horizontal and rectangular in shape. The lower base is 1215 ft. by 152 ft., and the upper base is 32 ft. wide. How many cubic yards of concrete were required?

Powder grain manufactured for the use of the large guns are small cylinders (composed of a nitrocellulose compound) with a number of longitudinal perforations. if each grain is 3/4 diameter, 1 3/4 in. long, and contains 7 cylindrical perforations 1/8 in. in diameter, find the amount of material in single grain.