jugoslav

https://web2.0calc.com/questions/question-help_7   (Very clever!!!) Bravo, MaxWong!!!

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BC => a = 6         AC => b = 8           AB => c = 10

∠A = tan^-1(6 / 8)             

∠ANC = 180 - (45 + ∠A)

By using the Law of Sines we can find side AN of triangle ACN.

AN ≈ 5.714           BN = 10 - AN ≈ 4.286

CN can be calculated by using this formula: CN = sqrt [ab(a + b + c)(a + b - c)] / a + b

CN ≈ 4.849 

By using the Intersecting chords theorem we can calculate the length of CM.

MN * CN = AN * BN             MN ≈ 5.05

CM = CN + MN ≈ 9.899  

What about XY = 18 ? 

XU = 4       VY = 8 

4 * 14 = 7 * 8 = 56           XU * UY = EU * UF

10 * 8 = 5 * 16 = 80         XV * VY = GV * VH

XY = 18 units

Yellow Δ area = √3        =>  3√3

Red   Δ area   = =>       => 25√3

White trapezoid area    => 33√3

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Total area = =>             => 61√3 square units

PC : AP = 1 : 2        MP : BP = 1 : 2

[MPC] = 1/12 [ABCD]           [BPC] = 1/6 [ABCD]

[APB] = 1/3 [ABCD]              [APMD] = 5/12 [ABCD]

[MPC] : [BPC] : [APB] : [APMD] = 1 : 2 : 4 : 5  

P (8.75, 9.75)       x + y = 18.5

First, I calculated the angle ADE (see my diagram), and then I subtracted that angle plus 90º angle (∠CDE) from 180, and I got 22.5

Value of 22.5 is the sum of angles BAO and AEC.

The ratio of these 2 angles is ∠BAO : ∠AEC = 2 : 1

And the rest is easy

∠CED = 90º       ∠ADE = 67.5º          ∠ACE = 90 + 67.5 = 157.5º

∠BAO + ∠AEC = 180 - 157.5 = 22.5º

∠BAO : ∠AEC = 2 : 1

Angle BAO = 15 degrees 

BD = 33 + 37

Angle CBD = 180 - (48 + 105)

[ACD] = 1/2 (AC * AD) = 20

[ABE] = 1/2 (AE * AB) = 27