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

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

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## Thursday, October 29, 2015

### Geometry Problem 1160: Triangle, Three Squares, Area, Equivalent Triangles

Labels:
area,
equivalent,
square,
triangle

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http://s16.postimg.org/u34puqelh/pro_1160.png

ReplyDeleteDenote S(XYZ)= area of XYZ

Draw altitudes AR and CL and lines per sketch

Observe that triangle B1HC congrurent to CLA ( case ASA) => B1H=CL

We have S(B1BC)= ½ .(BC1 x B1K).... (1)

But B1K= B1H+HK= CL+ BL

Replace this in (1) we have S(B1BC)=1/2. BC1 x(CL+BL)= S(ABC)+1/2. S(BLMC1)

Simillarly S(B2BA2)= S(ABC)+ ½. S(BRSA2)

Note that quadrilateral ACRL is cyclic => BL. BA=BR.BC => S(BLMC1)= S(BRSA2)

Replace it in above expressions we get S(B1BC)= S(B2BA2)

I doubt I would ever come up with such a brilliant proof like Peter's. I proved this with a somewhat dumber approach using algebra.

ReplyDeleteArea(BB1C1) = BB1*B1C1*sin(∠BB1C1)/2

Area(BB2A2) = BB2*A2B2*sin(∠BB2A2)/2

My approach involves these two steps:

(1) BB1*B1C1 == BB2*A2B2

(2) ∠BB1C1 == ∠BB2A2

They are not trivial but not too hard to reach either.