Let-ABC-be-a-triangle-and-h-a-the-altitude-through-A-Prove-that-b-c-2-a-2-4h-a-2-As-usual-a-b-c-denote-the-sides-BC-CA-AB-respectively-

Tinku Tara June 4, 2023 Geometry 0 Comments

Question Number 22079 by Tinkutara last updated on 10/Oct/17

Let ABC be a triangle and h_a the altitude through A. Prove that (b + c)^2 ≥ a^2 + 4h_a ^2 . (As usual a, b, c denote the sides BC, CA, AB respectively.)

$$\mathrm{Let}\:{ABC}\:\mathrm{be}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{and}\:{h}_{{a}} \:\mathrm{the} \\ $$$$\mathrm{altitude}\:\mathrm{through}\:{A}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left({b}\:+\:{c}\right)^{\mathrm{2}} \:\geqslant\:{a}^{\mathrm{2}} \:+\:\mathrm{4}{h}_{{a}} ^{\mathrm{2}} . \\ $$$$\left(\mathrm{As}\:\mathrm{usual}\:{a},\:{b},\:{c}\:\mathrm{denote}\:\mathrm{the}\:\mathrm{sides}\:{BC},\right. \\ $$$$\left.{CA},\:{AB}\:\mathrm{respectively}.\right) \\ $$

Commented by ajfour last updated on 11/Oct/17

$${solved}.\:{See}\:{Q}.\mathrm{22116} \\ $$

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Let-ABC-be-a-triangle-and-h-a-the-altitude-through-A-Prove-that-b-c-2-a-2-4h-a-2-As-usual-a-b-c-denote-the-sides-BC-CA-AB-respectively-

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