An interesting problem related to Pedal Triangle

Credits go to my lecturer for the problem, solution and the graph

The author is introducing a genuine way but I am introducing a conceptual way.

Two isosceles triangles are found for our calculations although right angle triangle and also triangle of various acute angles restricted to certain ranges are also found. For optimization, we prefer equilateral triangle but R = 5 and $\bigtriangleup$ = 10 do not allow for a validity. (24, 35, 121) of integer sides is the closest possible but unable to give exact area of 10. Criteria to satisfy is $\sin 2 A + \sin 2 B + \sin 2 C = \frac{10}{\frac12 5^2}$ = $\displaystyle \frac45.$

Formulas applied: $D = \sqrt{(x_2 -x_1)^2 + (y_2 - y_1)^2}$ and $\bigtriangleup$ = $\sqrt{s (s - a)(s - b)(s - c)}$ = $\frac12 |$ 4 points $|.$

With ABC of area of 10 with coordinates shown as follows, y-ordinate of M can be varied for obtaining a maximum area wanted. Initially, I put $\frac23$ of y-ordinate of A or B but there valid with a turning point for maximum with 5 as y-ordinate:

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A   -1.01042350456793   9.89684024054458
B   1.01042350456793    9.89684024054458
C   0   0
M   0   5.00000000000000 inside triangle ABC; chosen or preferred.　

    a = 49.7414320273969
    b = 49.7414320273969
    c = 10.1042350456793
    s = 54.7935495502366
    Area =  250


A   -4.24392501263905   2.35630930570602
B   4.24392501263905    2.35630930570602
C   0   0
M   0   5.00000000000000 outside triangle ABC; not chosen or not preferred.

    a = 24.2709152366882
    b = 24.2709152366882
    c = 42.4392501263905
    s = 45.4905402998835
    Area =  250

It is a replacement of equilateral triangle with isosceles triangle for finding such maximum. Otherwise, the span of location of M (x, y) is complicated to simulate even with random generator to be applied.

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a = 49.7414320273969
b = 49.7414320273969
c = 10.1042350456793

This is the case wanted. Its area is 250 (from data onto an area of only 10). Think it as an easy question can make a difficult question to become easy. Nevertheless, I am applying a reflective reliance instead of absolute way to answer this question. You may need to do further determination from my working described. [But my intuition makes me a correct answer to answer this question.] Don't ask me why. [(Ought to take back this sentence.)]

We are told that $10 \times$ $5^2$ = 250. More convincing to confirm and to remember for typical application:

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A   10  0
B   9.582575695 2
C   0   0
M   5   0

    a = 48.9453156465352
    b = 50.0000000000000
    c = 10.2154821844610
    s = 54.5803989154981
    Area =  250

For the right angled triangle above, M required is located at edge of half way along the hypotenuse.

Answer: $\boxed{> 250}$

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3 examples exceeded 250:

[1]
A = 90                      10.00000000000000   0
B = 78.2109107608991        9.582575695 2
C = 11.7890892391009        0   0
Sum =   0.799999999999999   10.00000000000000   0
Area =  9.99999999999999    9.58257569495584    2
a = 10.00000000000000       0   0
b = 9.78906312930703        10
c = 2.04309643689220        5.18    0
s = 10.9160797830996            
Area =  10          
a = 48.2            
b = 47.33563473         
c = 10.58323954         
s = 53.05943714         
Area =  250.3741202         

[2]
A = 86                      9.97564050259824    0
B = 82.3285707046987        9.705587526 2.004883796
C = 11.6714292953013        0   0
Sum =   0.800000000000000   9.97564050259824    0
Area =  9.99999999999999    9.70558752640135    2.004883796
a = 9.97564050259824        0   0
b = 9.91049889101209        10
c = 2.02298977903180        5   0.019
s = 10.9545645863211            
Area =  10          
a = 49.63556281         
b = 50.61761324         
c = 10.11502192         
s = 55.18409899         
Area =  251.0305482         

[3]
A = 87                      9.98629534754574    0
B = 81.3109739939435        9.680224906 2.002744692
C = 11.6890260060565        0   0
Sum =   0.800000000000003   9.98629534754574    0
Area =  10.00000000000000   9.68022490552900    2.002744692
a = 9.98629534754574        0   0
b = 9.88522840014036        10
c = 2.02599738815080        5.019   0
s = 10.9487605679184            
Area =  10          
a = 49.60487842         
b = 50.15036978         
c = 10.16848089         
s = 54.96186454         
Area =  251.9056644         

This is a first report to this question. Values > 250 as shown are $NOT$ maximized.