Geometry Problem. Post your solution in the comments box below.

Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to view more details of problem 1174.

## Saturday, December 26, 2015

### Geometry Problem 1174: Triangle, Quadrilateral, Double, Triple, Angle, Congruence, Excenter, Angle Bisector

Labels:
angle bisector,
congruence,
double angle,
excenter,
quadrilateral,
triangle,
triple

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Let the bisector of < ABC meet AC at E, CD at F and AD at G.

ReplyDeleteFor ease of typing I use @ for alpha and $ for theta

Easily < ABO = OBC = @/2 and < CBE = @ so that < EBD = 2@

Also since BG is the perpendicular bisector of AD, < BAE = EAF = BDE = EDF = $

So C is the incentre of Tr. ABF so < DFG = < AFG = < AFC = < CFE and each of these must therefore be essentially 60 degrees since the last 3 of these 4 angles add upto 180.

Hence from Tr. BDF, 2(&+$) = 60 and so

&+$ = 30.....(1)

In Tr. ABC, OA is an external angle bisector at A so x+@/2 = 90 -$/2 from which

x + (@+$)/2 = 90

Substituting @+& = 30 from (1) therefore

x+30/2 = 90 and hence x = 75

Sumith Peiris

Moratuwa

Sri Lanka

http://s17.postimg.org/59873mej3/pro_1174.png

ReplyDeleteLet angle bisector of angle ABD meet CD at F and AD at E

Observe that due to symmetry we have

∠(BAF)= ∠ (BDF)=2. Theta

And tri. AFD is isosceles

AC and BC are angles bisectors of ∠ (BAC) and ∠ (CFA)

So ∠ (BFC)= ∠ (CFA)=2.alpha+2.theta => ∠ (FDA)=alpha+theta

In right tri. BED we have alpha+theta=30

We have∠ (HOA)= theta/2 and ∠ (ABO)=alpha/2

In right tri. HBO we have x+theta/2+alpha/2= 90 => x= 75

Join AD and form the isosceles triangle ABD

ReplyDeleteE be foot of the perpendicular from B to AD

Say P be the point of intersection of BE and DC

Join AP and since ABD is isosceles m(CAP)=m(BAC)=θ and also APD is isosceles

=> C is the incenter of triangle ABP ----------(1)

A bit of angle chasing leads us to the below

m(BPC)=m(EPD)=2(α+θ) -----(2)

m(APC)=m(PAB)+m(PBA)=180-4(α+θ) -----------(3)

From (1), we have (2)=(3)

=>(α+θ)=30 ----------(4)

Now consider the triangle BAX

we have X=180-(90+θ/2)-α/2

=> X=75 degrees

Sorry for the typos

DeleteCorrected m(APC)=m(PAB)+m(PBA)=180-4(α+θ) -----------(3) to

"m(APC)=m(PAD)+m(PDA)=180-4(α+θ)" -----------(3)

Now consider the triangle BAX to "Now consider the triangle BAO"