Can you find area of the Blue triangle? | (Fun Geometry Problem) | #math #maths | #geometry

[juanalfaro]

I solved the problem by the tangent rule. [BCE] = BC*DE/2 = 4r/2 = 2r. To find r, let the middle angle in the figure be "b". Then tan (2a+b) = 12/r = [tan (a+b) + tan (a)] / [1-tan (a+b)*tan (a)] = [6/r+2/r] / [1-6/r*2/r] = [8/r] / [1-12/r^2] = 8r/ [r^2 - 12] = 12/r. Then 12(r^2-12) = 8r^2 --> 12r^2 - 144 = 8r^2 --> 4r^2 = 144 -> r^2 = 36 -> r=6m. Then [BCE] = 2*6 = 12 m^2

[joseluishablutzelaceijas]

Thank you very much for the problem and its solution. I think, an alternative solution would be as follows: As the angles AEB and CED are equal, one can exploit this by means of similarity. For doing so, one can extend EB in the direction from E to B to obtain the point "P" such that the angle APE is equal to 90°. Triangles APE and CDE are then similar and the points E, D, P and A all lie on the circle with center at the midpoint of AE and with radius |AE|/2. If one calls |EB| = m, one then has that |EB|*|BP| = |AB|*|BD| = 36 = m*|BP|, i.e. |BP| = 36/m. Furthermore, in triangle EBD one has by Pythagoras that |DE| = sqrt(m^2-36). By similarity of the triangles BAP and BED one has that |AP| = sqrt(m^2-36)*6/m and by similarity of the triangles EPA and EDC one has that sqrt(m^2-36)/2 = (m+36/m)/(6*sqrt(m^2-36)/m), which is equivalent to 3*m^2 - 108 = m^2 + 36, i.e. m^2 = 72 and sqrt(m^2-36) = 6 = |DE|. From this, it follows that [BCE] = |BC|*|DE|/2 = 4*6/2 = 12.

[west_park]

10:46 is a monstrous expression. could have been much simpler

[west_park]

10.23: please please simplify! divide the whole thing by -8=/=0 and use smaller numbers!!!

[west_park]

10:09 there should be no r in your equation. -8x+144x+5184=0. no r

[west_park]

you are using all small letters for everything, lentghs, areas, angles. this is so confusing. why not use greek letters for angles, and large letters for areas? or capital S with subscript for the respective triangle surface area. then we see immediately that y=3z, and y^2 = 9z^2. you can make it shorter. thanks

[matthieudutriaux]

area BCE = BC*DE/2=4*r/2=2*r

Let's call angle (BEC) = b

tan(a)=2/r
tan(a+b)=(4+2)/r=6/r
tan(2*a+b)=(6+4+2)/r=12/r


12/r=(2/r+6/r)/(1-2/r*6/r)
12/r=(8/r)/(1-12/r^2)
12=8/(1-12/r^2)
1-12/r^2=8/12=2/3
12/r^2=1-2/3
12/r^2=1/3
r^2=3*12=36
r=6

area BCE = 2*r = 2*6 = 12 m²

[west_park]

11:40 why so complicated: everyone knows that 288/-16 is negative and has no meaning. you are overcomplicating your explanation.