Let ABC be a triangle with incircle ω. The altitude from A to BC intersects ω at points E and F and intersects BC at point G so that F lies between E and G. Given that AE=5, EF=15, and FG=1, find the perimeter of ABC.
Answer: perimeter = 105
Please refer to this figure:
https://www.desmos.com/calculator/a9axs2u4pt
We first find the radius r and center O of omega, the incircle.
Setting BC along the x-axis with G at the origin, A=(0,21),
E=(0,16) and F=(0,1). Let's call H the point where omega
is tangent to BC. H = (x, 0)
Since EO = FO, center O lies along the line that
bisects EF. Call J the midpoint of EF. J = (0, 8.5)
Point O lies along y = 8.5, O = (x, 8.5)and omega has r = 8.5.
r = OE = OF = OH = 8.5, where here bold means length.
Considering the right triangle OJF, OF^2 = JF^2 + OJ^2
8.5 = sqrt(7.5^2 + x^2)
So x = 4 and O = (4, 8.5)
We next get P and Q, the points where omega is tangent
to AB and AC respectively. (Then from P and Q we can get the
coordinates for B and C and finally the perimeter.)
Setting PA^2 + PO^2 = AO^2 gives PA = 10.
Using this and the fact that APO is a right triangle
with and A and O known, we solve for P_x and P_y.
AO^2 - PO^2 = AP^2,
4^2 + (21-8.5)^2 - (P_x-4)^2-(P_y-8.5)^2 = P_x^2+(21-P_y)^2,
which gives: P = (-50/13, 153/-13) and a second solution
at (450/53, 833/53). (This second solution is Q)
The slope of line PA is m = (21-153/13) / (50/13) = 2.4
The line BA has the form y = 2.4 x + 21 which gives the
x intercept B = (-21/2.4, 0) = (-8.75, 0)
Since APO is congruent to AQO, with AQ = 10, the second
solution above is the location of Q = (Q_x, Q_y).
The slope of AC is
m =[(833/53)-21] / (450/53) = -28/45. So, AC is
y = -(28/45) x + 21 This gives for C, the x-intercept
C = (21/(28/45), 0) = (33.75, 0)
So, the full baseline BC = 33.75 - (-8.75) = 42.5
This gives the area of ABC = (1/2) (21) 42.5 = 446.25
Summarizing:
AB = sqrt ( 8.75^2 + 21^2 ) = 22.75
AC = sqrt (22.75^2 + 21^2 ) = 39.75
BC = 42.5
Summing gives perimeter = 105
We check this with the rule for incircles:
Area ABC = (1/2) r x perimeter
446.25 ?=? (1/2) 8.5 * 105
Yes
446.25 = 446.25 QED
Edited on January 9, 2025, 9:23 pm
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