Finding the height of a triangle without using trigonometry functions

Question

             A
            /|\
           / | \
          /  |  \
         /   |   \
        c    h    b 
       /     |     \
      /      |      \
  B  /___a1__|___a2__\ C

Given a triangle ABC as shown above. By using the Pythagoras theorem, a1= { c^2-b^2+(a1+a2)^2 } / 2(a1+a2) . How can I express h using a1+a2, b, and c?

Answer 1

By Heron's formula(It has algebraic proof using the Pythagorean theorem), we have $$ S = \sqrt{p(p-(a_1+a_2))(p-b)(p-c)}, \text{ where } p = \dfrac{a_1+a_2+b+c}{2}. $$ Also, $$ S = \dfrac{1}{2} (a_1+a_2)\times h $$ Can you proceed from here?

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$$ \dfrac{1}{2} a\times h = \sqrt{p(p-a)(p-b)(p-c)}\\ \dfrac{1}{4}a^2 h^2= p(p-a)(p-b)(p-c)\\ h^2=4p(p-a)(p-b)(p-c)/a^2 $$