Challenging Geometry Puzzle: Problem 1566. Share your solution by posting it in the comment box provided.

Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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We show that triangles ADC and BEC are congruent.

ReplyDeleteIndeed, the quadrilateral BEFD is concyclic because <EBD = external <DFC = 60º, then also <BEF = external <FDC = <ADC.

So triangles ADC and BEC have two equal angles <EBC = EBD = 60º = <ACD, and <ADC = <FDC = <BEF = <BEC. Also these triangles have congruent side AC = BC, both opposite to the equal angles <ADC = <BEC (then congruence by ASA). Observe that AD = EC, both opposite to the 60º angles, and finally CD = BE.

But AE + EB = AB = AC and CD = BE = EB implies AC = AE + CD, as desired.

Joaquim Maia

Rio de Janeiro

Brazil

BEFD is concyclic since < AFD = 60 = < B

ReplyDeleteSo < BEC = < ADC

Hence Triangles ADC & BEC are congruent ASA

Therefore AE = BD = BC - CD = AC - CD from which

AE + CD = AC

Sumith Peiris

Moratuwa

Sri Lanka

TrAEC,TrADB are congruent (AC=AC, <B=<B, <ACE=<BAD=<60°- <FAC) so

ReplyDeleteAE=BD thus AE+CD=BD+CD=BC=AC

Georgios Kousinioris

Gastouni

Greece

TrAEC,TrADB are congruent (AC=AB, <A=<B, <ACE=<BAD=<60°-<FAC) so

ReplyDeleteAE=BD

Therefore AE+CD=BD+CD=BC=AC

Georgios Kousinioris

Gastouni

Greece