Given a sentence in Tagalog language, translate the sentence to English language keeping the meaning of the original sentence intact.

Tagalog sentence: Ang point M ay ang midpoint ng Arc BC sa circle c. Kaya, ang dalawang Chords BM at CM ng c bilog ay may pantay na haba. Kaya, ang chordal angles ng mga dalawang Chords ay dapat pantay-pantay, sa ibang salita, mayroon kaming &lt; BAM = &lt; CAM. Kaya, ang point M kasinungalingan sa anggulo panggitnang guhit ng anggulo BAC.
The point M is the midpoint of the arc BC on the circle c. Thus, the two chords BM and CM of the circle c have equal length. Hence, the chordal angles of these two chords must also be equal; in other words, we have &lt; BAM = &lt; CAM. Hence, the point M lies on the angle bisector of the angle BAC. Similarly, the point N lies on the angle bisector of the angle BAD. But, of course, the angle bisector of the angle BAD is the exterior angle bisector of the angle BAC; hence, since the interior and the exterior angle bisector of an angle are always perpendicular to each other, the angle bisector of the angle BAC and the angle bisector of the angle BAD are perpendicular to each other. Hence, &lt; MAN = 90°.

Tagalog sentence: Ang kawalan ng kaalaman ng mga tauhan ng gawa ng kalendaryo ng tumpak na mas mababa kaysa sa mga ito ay dapat ngunit ito ay nagmula sa paggamit sa 1075 at lasted tungkol sa 20 taon bago na papalitan.
The inefficiency of the staff made the calendar less accurate than it should have been but it came into use in 1075 and lasted about 20 years before being replaced.

Tagalog sentence: Ang enquiries kung saan ang mga dakilang gawa ay batay marahil ay natupad sa kalakhan sa Assos at Lesbos.

The enquiries upon which those great works were based were probably carried out largely in Assos and Lesbos.