×

TWO OF MY FORMULAS !!!!!!!!!!!!!!!!!!!

Hello, my dear brilliant members. Well, I wanted to give you some simple challenges. These are 2 of my new formulas which are (A)Finding the height of any triangle with sides a,b,c in terms of a, b and c. Obviously, there can be 3 heights but you find for base b. (B) Finding the length of median of any triangle of side a, b, c in terms of a, b and c and the median should be bisecting side b. Well, it is extremely easy and the challenge is that you find these formulas with proper proofs. Don't Google it as I am pretty sure you all can do it. Thank you and everyone comment on this, try and find the formulas yourself and post it in the comments.

Note by Kushagra Sahni
3 years, 2 months ago

Sort by:

u suck · 2 years ago

Proofs: (A) Take $$\triangle ABC$$ with an altitude $$BD$$ of length $$h$$ perpendicular to $$b$$. Let $$DC=x$$, and thus, $$AD=b-x$$. Now, by Pythagoras Theorem, $$a^{2}-x^{2}=c^{2}-(b-x)^{2}$$ $$\Rightarrow$$ $$a^{2}+b^{2}-c^{2}=2bx$$ $$\Rightarrow$$ $$x=\frac{a^{2}+b^{2}-c^{2}}{2b}$$

Also, $$x^{2}+h^{2}=a^{2}$$ $$\Rightarrow$$ $$h^{2}=a^{2}+x^{2}$$

Substituting for $$x$$ into the first equation, $$h^{2}=c^{2}-(b-\frac{a^{2}+b^{2}+c^{2}}{2b})^{2}$$

By simplifying and bringing to the same denominator, we get $$h=\sqrt{\frac{2b^2c^2 + 2c^2b^2 + 2a^2b^2-a^4-b^4-c^4}{4b^2}}$$ $$\Rightarrow$$ $$h=\frac{\sqrt{2b^2c^2+2c^2a^2+2a^2b^2-a^4-b^4-c^4}}{2b}$$, which can be simplified, as shown in my other comment. · 3 years, 2 months ago

It still can be simplified. Apply the identity a^2-b^2 · 3 years, 2 months ago

And yes. Post a proof for median one as well · 3 years, 2 months ago

I have gotten both formulae.

(A) $$h_b = \frac{\sqrt{2a^{2}b^{2}+2b^{2}c^{2}+2c^{2}a^{2}-a^{4}-b^{4}-c^{4}}}{2b}$$, where $$h_b$$ denotes the altitude incident on $$b$$

(B) $$x_b = \frac{\sqrt{2a^{2}+2c^{2}-b^{2}}}{2}$$, where $$x_b$$ denotes the median incident on $$b$$

Should I post a proof? · 3 years, 2 months ago

Sorry, been caught up with IIT classes · 3 years, 2 months ago

No problem friend, please reshare this note and simplify your altitude formula and post a proof for both. · 3 years, 2 months ago

AND YES, ALSO TELL ME THAT ARE THESE GOOD DISCOVERIES OR VERY EASY ONES ? · 3 years, 2 months ago

YOUR MEDIAN ONE IS CORRECT, BUT THE ALTITUDE ONE CAN BE SIMPLIFIED FURTHER. SIMPLIFY IT USING SUITABLE IDENTITIES AND THEN POST A PROOF FOR BOTH · 3 years, 2 months ago

The altitude formula can be simplified to $$h_b=\frac{\sqrt{(a^{2}+b^{2}+c^{2})^{2}-4(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2})}}{2b}$$ · 3 years, 2 months ago