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Question Number 54068 by ajfour last updated on 28/Jan/19

Commented by ajfour last updated on 28/Jan/19

locate P (x,y) such that △APB, △BPC, and △CPA each have the same perimeter. (in terms of a,b,c)

locateP(x,y)suchthat△APB,△BPC,and△CPAeachhavethesameperimeter.(intermsofa,b,c)

Commented by mr W last updated on 28/Jan/19

this is (not always) possible.

thisis(notalways)possible.

Answered by mr W last updated on 28/Jan/19

s=(√((x−x_A )^2 +(y−y_A )^2 ))+(√((x−x_B )^2 +(y−y_B )^2 ))+(√((x_A −x_B )^2 +(y_A −y_B )^2 )) s=(√((x−x_B )^2 +(y−y_B )^2 ))+(√((x−x_C )^2 +(y−y_C )^2 ))+(√((x_B −x_C )^2 +(y_B −y_C )^2 )) s=(√((x−x_C )^2 +(y−y_C )^2 ))+(√((x−x_A )^2 +(y−y_A )^2 ))+(√((x_C −x_A )^2 +(y_C −y_A )^2 )) 3 eqn. with 3 unknowns x,y,s. A(0,0) B(b,0) C(c,d) s=(√(x^2 +y^2 ))+(√((x−b)^2 +y^2 ))+b s=(√((x−b)^2 +y^2 ))+(√((x−c)^2 +(y−d)^2 ))+(√((b−c)^2 +d^2 )) s=(√((x−c)^2 +(y−d)^2 ))+(√(x^2 +y^2 ))+(√(c^2 +d^2 )) ⇒(√(x^2 +y^2 ))−(√((x−c)^2 +(y−d)^2 ))=(√((b−c)^2 +d^2 ))−b ⇒(√((x−b)^2 +y^2 ))−(√(x^2 +y^2 ))=(√(c^2 +d^2 ))−(√((b−c)^2 +d^2 )) example: A(0,0), B(5,0), C(2,4) ⇒P(2.7888,1.1056)

s=(x−xA)2+(y−yA)2+(x−xB)2+(y−yB)2+(xA−xB)2+(yA−yB)2s=(x−xB)2+(y−yB)2+(x−xC)2+(y−yC)2+(xB−xC)2+(yB−yC)2s=(x−xC)2+(y−yC)2+(x−xA)2+(y−yA)2+(xC−xA)2+(yC−yA)23eqn.with3unknownsx,y,s.A(0,0)B(b,0)C(c,d)s=x2+y2+(x−b)2+y2+bs=(x−b)2+y2+(x−c)2+(y−d)2+(b−c)2+d2s=(x−c)2+(y−d)2+x2+y2+c2+d2⇒x2+y2−(x−c)2+(y−d)2=(b−c)2+d2−b⇒(x−b)2+y2−x2+y2=c2+d2−(b−c)2+d2example:A(0,0),B(5,0),C(2,4)⇒P(2.7888,1.1056)

Commented by mr W last updated on 28/Jan/19

that was an error.

thatwasanerror.

Commented by ajfour last updated on 28/Jan/19

Thank you Sir .

ThankyouSir.

Commented by ajfour last updated on 28/Jan/19

you are seldom in error Sir.

youareseldominerrorSir.

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