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Math Help - Derivation of the Parabola's Focal Property Using a Sphere Withing a Cone

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    Derivation of the Parabola's Focal Property Using a Sphere Withing a Cone

    KMaL - Rita Ks: Conics and Dandelin spheres


    I don't understand how we conclude "The angles P*PP' and P*PD are equal to the half of the apex angle of the cone"

    The proof is using AAS, correct? I see that the triangles will share a leg and a right angle, but I don't understand how we can get another angle equal to use AAS.
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    Quote Originally Posted by zg12 View Post
    KMaL - Rita Ks: Conics and Dandelin spheres


    I don't understand how we conclude "The angles P*PP' and P*PD are equal to the half of the apex angle of the cone"

    The proof is using AAS, correct? I see that the triangles will share a leg and a right angle, but I don't understand how we can get another angle equal to use AAS.
    1. I've modified the right sketch a little bit by adding the axis of the cone which passes through the center of the circle k and 2 axes to indicate the horizontal plane which contains the circle k.

    2. Therefore \overline{PP^*} \parallel axis_{cone}

    3. \angle(PP'Q) = \frac12(angle\ of\ cone's\ apex)

    4. \angle(P'PP^*) = \angle(PP'Q)
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