Question 966259
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Let &nbsp;<B>ABCD</B>&nbsp; be a trapezoid with the bases &nbsp;<B>AB</B>&nbsp; and &nbsp;<B>DC</B>, &nbsp;and &nbsp;<B>AC</B>&nbsp; and &nbsp;<B>BD</B>&nbsp; be its diagonals.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 

(<B>Figure 1</B>). &nbsp;We need to prove that the triangles &nbsp;{{{DELTA}}}<B>ACD</B>&nbsp; and &nbsp;{{{DELTA}}}<B>BCD</B>&nbsp; have equal areas. 


Let us draw the altitudes &nbsp;<B>AE</B>&nbsp; and &nbsp;<B>BF</B>&nbsp; in the trapezoid &nbsp;<B>ABCD</B>&nbsp; from vertices &nbsp;<B>A</B>&nbsp; and &nbsp;<B>B</B>&nbsp; 

to the base &nbsp;<B>DC</B>&nbsp; (<B>Figure 1</B>). 

Notice that these altitudes are congruent as the opposite sides of the rectangular &nbsp;<B>ABFE</B>. 


Now, &nbsp;the triangles &nbsp;{{{DELTA}}}<B>ACD</B>&nbsp; and &nbsp;{{{DELTA}}}<B>BCD</B>&nbsp; share the common base &nbsp;<B>DC</B>&nbsp; and have 

congruent altitudes to the base. &nbsp;Therefore, &nbsp;these triangles have the same area.

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{{{drawing( 250, 100,  -0.5, 9.5, -0.5, 3.5, 
            line( 0.0,  0.0, 9.0,  0.0), 
            line( 0.0,  0.0, 3.0,  3.0),
            line( 3.0,  3.0, 8.0,  3.0),
            line( 9.0,  0.0, 8.0,  3.0),

            locate( -0.1,  0.0, D),
            locate(  8.9,  0.0, C),
            locate(  2.8,  3.6, A),
            locate(  8.0,  3.6, B),

       blue(line( 0.0, 0.0, 8.0, 3.0)),
       blue(line( 3.0, 3.0, 9.0, 0.0)),

        red(line(3,0, 3.0, 3.0, 0.0)),
        red(line(8.0, 3.0, 8.0, 0.0)),

            locate(  2.8,  0.0, E),
            locate(  7.8,  0.0, F)
)}}}


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>Figure 1</B>
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