Saturday, May 23, 2009

Problem 291: Triangle, Circle, Radius, Perpendicular

Proposed Problem
Click the figure below to see the complete problem 291.

 Problem 291: Triangle, Circle, Radius, Perpendicular.
See also:
Complete Problem 291
Collection of Geometry Problems

Level: High School, SAT Prep, College geometry


  1. 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
    However, 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:

  2. 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)

    1. Correction: Let O be the circum-centre of Tr. ABC.

  3. Apply sine law in triangle ABC=> sin(B)= b/2R
    Apply 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

  4. BEDF is cyclical and BD is the diameter of the circle.
    Let 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:


  5. Sumith Peiris' answer:

    Drop 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
    Sri Lanka

  6. 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