Proposed Problem
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Complete Problem 291
Collection of Geometry Problems
Level: High School, SAT Prep, College geometry
Saturday, May 23, 2009
Problem 291: Triangle, Circle, Radius, Perpendicular
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In quad. DFBE we've DF & DE perpendicular BF & BE respectively. Hence DFBE is concyclic. Therefore, angle ABC = angle EDF = let's say, B. Now let angle EDB = B1 & angle BDF = B2. Hence, sin(B) = sin(B1+B2) = sinB1*cosB2 + oosB1*sinB2 = (BE/d)(DF/d) + (DE/d)(BF/d) = (BE*DF+DE*BF)/d^2
ReplyDeleteHowever, by Ptolemy's Theorem, (BE*DF+DE*BF =d*e
Thus, sin(B)= d*e/d^2 = e/d ---------(1)
Now area of Tr. ABC = acsin(B)/2. Hence, it's circumradius R = abc/4(acsin(B)/2) = b/2sin(B) or R = b/2(e/d) using equation (1). This gives:
2R*e = b*d
Vihaan Uplenchwar, Dubai: vihaanup@gmail.com
Let M be the mid-point of BD & O the circumcircle of Tr. ABC. Then clearly, FM=DM=d/2 and /_FMB = 2*/_FDE=2*/_B since F,D,E & B are concyclic. Further, Tr. FME is isosceles. Likewise, Tr. OAC is isosceles with an apex /_ of 2*B. Hence, Tr. FME /// Tr. OAC which gives us, FM/FE = OA/AC or d/(2e) = R/b or R = bd/(2e)
ReplyDeleteCorrection: Let O be the circum-centre of Tr. ABC.
DeleteApply sine law in triangle ABC=> sin(B)= b/2R
ReplyDeleteApply sine law in triangle EDF => sin(EDF)= e/d
Since EDFB is cyclic quadrilateral so angle (B)= angle (EDF)
So we have b/2R= e/d => 2R.e=bd
BEDF is cyclical and BD is the diameter of the circle.
ReplyDeleteLet M be the midpoint of BD then M is the center of the circle BEDF.
Let N be the center of the circumcircle of ABC.
Draw the circle of M.
Join FM and extend FM to meet the circle M in the point P.
Join P with E.
Join AN and extend AN to meet the circle N in the point Q.
Join Q with C
We can see that ∠ABC=∠FDE and ∠ABC =∠AQC.
Also ∠FDE =∠FPE therefore ∠ABC=∠FPE
Because FPE and AQC are right angled triangles they are similar and from similarity we get:
AC/AQ=FE/FP
b/2R=e/d
bd=2Re
Sumith Peiris' answer:
ReplyDeleteDrop perpendiculars FP (=p) to BC and OQ to AC. Let FB = q
From similar Triangles,
b/2R = p/q. and e/d = p/q and the result follows
Sumith Peiris
Moratuwa
Sri Lanka
BEDF are concyclic and <EDF = <ABC therefore e, the chord of <EDF in circle BEDF, is to b, the chord of <ABC in circle ABC, what the radius of BEDF is to R i.e. e/b = (d/2)/R QED
ReplyDelete