How to prove the formula of the relationship between the radius of the inscribed circle and the trilateral of a right triangle?

How to prove the formula of the relationship between the radius of the inscribed circle and the trilateral of a right triangle?

It is known that in RT △ ABC ∠ C = 90 °, the inscribed circle ⊙ o is cut into AB, BC and Ca respectively in D, e and F
Verification: ⊙ o radius = (a + B-C) / 2
Prove that ∵ o cuts AB, BC and Ca at points D, e and F,
According to the tangent length theorem, AE = AF, BD = BF, AC + bc-ab = AE + CE + BD + cd-af-bf = CD + CE
∵ in quadrilateral CDOE, ∠ C = ∠ CDO = ∠ CEO = 90 ° and OD = OE,
The quadrilateral CDOE is a square, CD = CE = OD,
⊙ o radius od = CD = (AC + bc-ab) / 2 = (a + B-C) / 2, certificate completed

There are two formulas for the radius of the inscribed circle of a right triangle. How can we push each other? (1)r=a+b-c/2 (2)r=ab/a+b+c

(1) (a+b-c)/2=｛〔(a+b-c)/2〕*(a+b+c)｝/(a+b+c)
=｛〔(a+b)^2-c^2〕/2｝/(a+b+c)
Because (a + b) ^ 2 = a ^ 2 + B ^ 2 + 2Ab = C ^ 2 + 2Ab,
So (a + B-C) / 2 = {[C ^ 2 + 2ab-c ^ 2] / 2} / (a + B + C)
=ab/(a+b+c )
(2)r=ab/(a+b+c) =ab(a+b-c)/〔(a+b+c)(a+b-c)〕
=ab(a+b-c)/〔(a+b)^2-c^2〕
=ab(a+b-c)/2ab
=(a+b-c)/2

Programming in C language, the three side lengths a, B and C of the triangle are known, and the formula for calculating the triangle area is: S = 1 / 2 (a + B + C), area = under the root sign [S (S-A) (S-B) (S-C)] It is required to write a program, input the values of a, B and C from the keyboard, calculate and output the area of the triangle [tip: when running the program, ensure that the input values of a, B and C meet the conditions for the establishment of the triangle, so that the calculated triangle area is meaningful. In addition, write the mathematical formula of area calculation into a legal C language expression as follows: area = sqrt(s*(s-a)*(s-b)*(s-c)) Note that it is written as: area = sqrt(s(s-a)(s-b)(s-c)) It is not wrong =, write the following C language expression: s = 0.5*(a+b+c)

#Include#includevoid main() {float a, B, C, s, area; printf ("enter a, B, C in sequence (spaces identify a number):"); scanf ("% F% F,", & A, & B, & C); s = (float) 0.5 * (a + B + C); area = (float) sqrt (s * (S-A) * (S-B) * (S-C)); printf ("area is:% F", area

Given that the area of △ ABC is s and the lengths of the three sides are a, B and C respectively, the radius of the inscribed circle of △ ABC is equal to _

Let the center of the inscribed circle be I, and the tangent points of the inscribed circle and AB, BC and Ca are f, D and e respectively, connecting AI, Bi, CI, Di, EI and fi. Then ID, ie and if are the heights of △ IBC, △ ICA and △ IAB respectively, and id = ie = if = R (R is the radius of the inscribed circle). ‡ s △ IBC = 12bc • id = 12ar, s △ ICA = 12ca • ie =

What is the area formula of right triangle

Multiply the two right angle sides and divide by 2

Formula of inscribed circle radius of right triangle

Radius of inscribed circle of right triangle r = 1 / 2 (AB + ac-bc) (formula 1) r = AB * AC / (AB + triangle with BC as oblique edge 1. R = 1 / 2 (AB + ac-bc) (formula 1) uses the property of tangent