Online Geometry theorems, problems, solutions, and related topics.
See also:Complete Problem 112, Collection of Geometry ProblemsArea of Square and Triangle. Level: High School, SAT Prep, College geometryPost your solutions or ideas in the comments.
You need additional points to prove this.G - the center of semicircleThen AG = DG = EG = FG = ½√SAnd FG is perpendicular to CFSo knowing that CG = ½√(5S) lets us compute CF = √S = AB = AD = BC = CDUsing analytic coordinates, we can use the equations for CF = √S and FG = ½√S to prove that F is 1/5 of the way from A to D and 2/5 the way from A to B. Alternately, we can add the point H which is the closest point on AD to F, and use the similar 1:2:√5 triangles to prove the same idea.Thus DE = √(S/2), DF = √(4S/5), and EF = √(S/10) which gives us the area of S/10
If we take the midpoint of AC as the origin and label A:(-a,0),B:(-a,2a),C:(a,2a)& D:(a,0) then E is clearly (0,a). Now CF & CD are both tangents to the semicircle so CF = CD = 2a which means F is one of the intersections of the two circles given by x^2+y^2=a^2 &(x-a)^2+(y-2a)^2= 4a^2. This gives F as (-3a/5,4a/5). Therefore, Tr. DEF =S1=(1/2)[a(a-4a/5)-(3a/5(-a)] = 4a^2/10But S = 2a*2a = 4a^2; hence S1 = S/10Ajit: email@example.com
I tried a solution without coordinates in problem 112.As triangles DEF and BEF have equal bases BE and DE and same altitude relative to that bases, then both have area S1. Being O the center of the semicircle, and s the side of the square ABCD, let’s write the altitude of triangle BAF as FG = p, the altitude of triangle AFD as FH = q, and ang(ADF) = a.DOF is isosceles then ang(DOF) = 180º - 2a. Quadrilateral CDOF is cyclic, so ang(DCF) = 2a. We have CD = CF (tangents from C to the circle) thus CDOF is a kite, OC bisects the angle DCF and ang(DCO) = a.We can see now that FHD and ODC are similar, with q/(s-p) = (s/2)/s so p = s - 2q (1).Also AHF and AFD are similar, because ang(AFD) = 90º, so AHF and ODC are similar, with p/q = (s/2)/s and q = 2p (2).From (1) and (2) we get p = s/5 and q = 2s/5.Let S2 and S3 be the areas of AFD and AFB respectively. We have, successively 2S1 + S2 + S3 = S/2, then 2S1 + sq/2 + sp/2 = S/2, then 2S1 + (s^2)/5 + (s^2)/10 = S/2, then 2S1 + (3S)/10 = S/2, then S1 = S/10.
Let FD, OC meet at XThen AFD, FXC,DXC are all congruent triangles and since each are similar to Tr.ODC, OC = sqrt5 a/ 2 where a = side of the squareHence XD = a^2/4 /((sqrt5 a/2)/2) = a/sqrt5 = AFS(AFD) = S(ODC) AF^2/OD ^2 = a^2/5S(FCD) = 2a^2/5So d the altitude from F in Tr. AFD = a^/5/(a/2) = 2a/5So S(BFC) = (1/2)(3a/5)a = 3a^2/10Hence S(BFD) = 2a^2/5+3a^2/10-a^2/2 = a^2/5Therefore S(FED) = S(BFD)/2 = a^2/10 = S/10Sumith PeirisMoratuwaSri Lanka