G.25 base angles of an isosceles triangle — Heath
I.5
A
B
C
F
G
D
E
Let < A B C > be an isosceles triangle having the side / A B / equal to the side / A C / ;
and let the straight lines / B D / , / C E / be produced further in a straight line with / A B / , / A C / . [G.18]
I say that the angle < A B C ) is equal to the angle < A C B ) , and the angle < C B D ) to the angle < B C E ) .
Let a point F be taken at random on / B D / ; from / A E / the greater let / A G / be cut off equal to / A F / the less; [G.21]
and let the straight lines / F C / , / G B / be joined. [G.14]
Then, since / A F / is equal to / A G / and / A B / to / A C / ,
the two sides / F A / , / A C / are equal to the two sides / G A / , / A B / , respectively;
and they contain a common angle, the angle < F A G ) .
Therefore the base / F C / is equal to the base / G B / ,
and the triangle < A F C > is equal to the triangle < A G B > , and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend,
that is, the angle < A C F ) to the angle < A B G ) ,
[G.24] and the angle
< A F C ) to the angle
< A G B ) .
And, since the whole / A F / is equal to the whole / A G / , and in these / A B / is equal to / A C / ,
the remainder / B F / is equal to the remainder / C G / .
But / F C / was also proved equal to / G B / ; therefore the two sides / B F / , / F C / are equal to the two sides / C G / , / G B / respectively; and the angle < B F C ) is equal to the angle < C G B ) ,
while the base / B C / is common to them;
therefore the triangle < B F C > is also equal to the triangle < C G B > , and the remaining angles will be equal to the remaining
angles respectively, namely those which the equal sides subtend;
therefore the angle < F B C ) is equal to the angle < G C B ) ,
and the angle < B C F ) to the angle < C B G ) .
Accordingly, since the whole angle < A B G ) was proved equal to the angle < A C F ) , and in these the angle < C B G ) is equal to the angle < B C F ) , the remaining angle < A B C ) is equal to the remaining angle < A C B ) ;
and they are at the base of the triangle < A B C > .
But the angle < F B C ) was also proved equal to the angle < G C B ) ;
and they are under the base.
Therefore etc. Q. E. D.
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"G.2" [label="G.2", tooltip="G.2 (I.def.2): A line [ A B ] is a projection of a relationship of two points A, B.", fillcolor="#224422", URL="../G.2/index.html", target="_top"];
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"G.6" [label="G.6", tooltip="G.6 (I.def.6): The extremities of a surface are lines or arcs.", fillcolor="#224422", URL="../G.6/index.html", target="_top"];
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"G.12" [label="G.12", tooltip="G.12 (I.def.19): Rectilineal figures are contained by segments: trilateral by three, quadrilateral by four, and multilateral by more than four.", fillcolor="#224422", URL="../G.12/index.html", target="_top"];
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"G.16" [label="G.16", tooltip="G.16 (I.cn.1): Things equal to the same thing are equal to one another.", fillcolor="#442222", URL="../G.16/index.html", target="_top"];
"G.17" [label="G.17", tooltip="G.17 (I.1): Construct an equilateral triangle on a given segment.", fillcolor="#222244", URL="../G.17/index.html", target="_top"];
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"G.22" [label="G.22", tooltip="G.22 (I.def.8): A plane angle is the inclination to one another of two lines or arcs in a plane that meet and do not lie on the same line.", fillcolor="#224422", URL="../G.22/index.html", target="_top"];
"G.23" [label="G.23", tooltip="G.23 (I.cn.4): Things / A B /, / D E / that coincide with one another are equal to one another.", fillcolor="#442222", URL="../G.23/index.html", target="_top"];
"G.24" [label="G.24", tooltip="G.24 (I.4): Triangles with two sides and the included angle equal are congruent (SAS).", fillcolor="#222244", URL="../G.24/index.html", target="_top"];
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"G.2" -> "G.1";
"G.3" -> "G.1";
"G.4" -> "G.1";
"G.10" -> "G.1";
"G.11" -> "G.1";
"G.14" -> "G.1";
"G.15" -> "G.1";
"G.17" -> "G.1";
"G.20" -> "G.1";
"G.21" -> "G.1";
"G.24" -> "G.1";
"G.25" -> "G.1";
"G.3" -> "G.2";
"G.4" -> "G.2";
"G.6" -> "G.2";
"G.7" -> "G.2";
"G.18" -> "G.2";
"G.22" -> "G.2";
"G.4" -> "G.3";
"G.6" -> "G.3";
"G.8" -> "G.3";
"G.20" -> "G.3";
"G.7" -> "G.4";
"G.10" -> "G.4";
"G.11" -> "G.4";
"G.12" -> "G.4";
"G.14" -> "G.4";
"G.17" -> "G.4";
"G.18" -> "G.4";
"G.20" -> "G.4";
"G.21" -> "G.4";
"G.22" -> "G.4";
"G.24" -> "G.4";
"G.25" -> "G.4";
"G.6" -> "G.5";
"G.7" -> "G.5";
"G.10" -> "G.6";
"G.10" -> "G.7";
"G.22" -> "G.7";
"G.9" -> "G.8";
"G.10" -> "G.9";
"G.12" -> "G.9";
"G.11" -> "G.10";
"G.15" -> "G.10";
"G.17" -> "G.10";
"G.20" -> "G.10";
"G.21" -> "G.10";
"G.15" -> "G.11";
"G.17" -> "G.11";
"G.20" -> "G.11";
"G.21" -> "G.11";
"G.13" -> "G.12";
"G.17" -> "G.12";
"G.20" -> "G.12";
"G.24" -> "G.12";
"G.25" -> "G.12";
"G.17" -> "G.13";
"G.25" -> "G.13";
"G.17" -> "G.14";
"G.20" -> "G.14";
"G.25" -> "G.14";
"G.17" -> "G.15";
"G.20" -> "G.15";
"G.21" -> "G.15";
"G.17" -> "G.16";
"G.20" -> "G.16";
"G.21" -> "G.16";
"G.20" -> "G.17";
"G.20" -> "G.18";
"G.25" -> "G.18";
"G.20" -> "G.19";
"G.25" -> "G.19";
"G.21" -> "G.20";
"G.25" -> "G.21";
"G.24" -> "G.22";
"G.25" -> "G.22";
"G.24" -> "G.23";
"G.25" -> "G.24";
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Required for
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