Kitap 11
35
But GL is perpendicular to the plane through BA, AC; therefore HK is also perpendicular to the plane through. BA, AC. [XI. 8]
35
From the points K, N let KC, NF, KB, NE be drawn perpendicular to the straight lines AC, DF, AB, DE, and let HC, CB, MF, FE be joined.
35
Since the square on HA is equal to the squares on HK, KA, and the squares on KC, CA are equal to the square on KA, [I. 47] therefore the square on HA is also equal to the squares on HK, KC, CA.
35
But the square on HC is equal to the squares on HK, KC; [I. 47] therefore the square on HA is equal to the squares on HC, CA.
35
Therefore the angle HCA is right. [I. 48]
35
For the same reason the angle DFM is also right.
35
Therefore the angle ACH is equal to the angle DFM.
35
But the angle HAC is also equal to the angle MDF.
35
Therefore MDF, HAC are two triangles which have two angles equal to two angles respectively, and one side equal to one side, namely, that subtending one of the equal angles, that is, HA equal to MD; therefore they will also have the remaining sides equal to the remaining sides respectively. [I. 26]
35
Therefore AC is equal to DF.
35
Similarly we can prove that AB is also equal to DE.
35
Since then AC is equal to DF, and AB to DE, the two sides CA, AB are equal to the two sides FD, DE.
35
But the angle CAB is also equal to the angle FDE; therefore the base BC is equal to the base EF, the triangle to the triangle, and the remaining angles to the remaining angles; [I. 4] therefore the angle ACB is equal to the angle DFE.
35
But the right angle ACK is also equal to the right angle DFN; therefore the remaining angle BCK is also equal to the remaining angle EFN.
35
For the same reason the angle CBK is also equal to the angle FEN.
35
Therefore BCK, EFN are two triangles which have two angles equal to two angles respectively, and one side equal to one side, namely, that adjacent to the equal angles, that is, BC equal to EF; therefore they will also have the remaining sides equal to the remaining sides. [I. 26]
35
Therefore CK is equal to FN.
