See complete Problem 41

Sangaku, Mickey Mouse, Circles, Tangent. Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

## Monday, May 19, 2008

### Elearn Geometry Problem 41

Labels:
circle,
mickey mouse,
sangaku,
tangent

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Using Pythagoras, BC^2 = d^2 + (c - b)^2

ReplyDeleteLet angle CAB = z

Using cosine rule: BC^2 = (a+c)^2 + (a+b)^2 - 2(a+c)(a+b)cos(z)

Or d^2 + (c - b)^2 = (a+c)^2 + ((a+b)^2 - 2(a+c)(a+b)cos(z)

From where, d^2+ b^2+c^2 - 2bc = a^2+c^2+2ac + a^2+b^2+2ab - 2(a+c)(a+b) cos(z)

Or d^2=2a^2+2ac+2ab+2ac-2(a+c)(a+b)cos(z)

Or d^2=2(a+b)(a+c)-2(a+c)(a+b)cos(z) or d^2

= 2(a+b)(a+c)(1-cos(z) --------(i)

From the triangle in circle A we can write,

x^2 =a^2 +a^2-2a^2*cos(z)

=2a^2(1-cos(z)

Or(1-cos(z)) = x^2/2a^2-------------(ii)

From (i) and (ii), we get:

d^2 = 2(a+b)(a+c)x^2/2a^2

= (a+b)(a+c)x^2/a^2

Or x^2 = a^2*d^2/[(a+b)(a+c)]

QED

Ajit:ajitathle@gmail.com

By cosine rule,

ReplyDeleted^2 +(b-c)^2 =BC^2

=(a+b)^2+(a+c)^2-2(a+b)(a+c)cos(ang(BAC))

=(a+b)^2+(a+c)^2-2(a+b)(a+c)(a^2+a^2-x^2)/2aa

=2a^2+b^2+c^2+2ab+2ac-2(a^2+ab+ac+bc)

+(a+b)(a+c)(x^2/a^2)

<=>

d^2=(a+b)(a+c)(x^2/a^2).

We get, x^2 =(a^2 d^2)/(a+b)(a+c)

From Moscow: I have pure synthetic solution but not sure if correct. Please check. Dropping perpendiculars from A, F, and G onto DE to get A' F' G', we have 1)A'F'/A'D=a/(a+b)and 2)A'G'/A'E=a/(a+c). Looking at pentagon ABCDE, We note that angle A + angle B + angle C + 90 + 90= 540, so angles A, B, and C add to 360. We see that angle FDE is half of B and angle GED is half of C. Therefore FD and GE meet at point P such that angle FPG is half of angle A so P is on circle A. But angle BDF = 90-angle FDE = BFD = AFP = APF, so AP is perpendicular to DE. x/d=PF/PE=PG/PD, so x^2/d^2=(PF*PG)/(PE*PD)=(A'F'*A'G')/(A'D*A'E)=a^2/((a+b)(a+c)). The last two equality statements we get from 1) and 2) and we are done.

ReplyDeleteTo Ivan from Moscow, Problem 41. Your great pure synthetic solution is correct. To complete your solution, what is the reason for the step: x/d=PF/PE=PG/PD. Thanks.

Delete